## 17Calculus Precalculus - Cramer's Rule

This page covers how to solve linear systems of equations using determinants.

We can use matrices and determinants to solve systems of linear equations using a technique called Cramers Rule. We will show how to do this with 2 equations and 2 unknowns. However, the concept can be extended to higher order systems.

Problem

We are given a system of linear equations of the form
$$\begin{array}{ccccc} ax & + & by & = & z_0 \\ cx & + & dy & = & z_1 \end{array}$$
where $$a, b, c, d, z_0, z_1$$ are all real constants and the variables are $$x$$ and $$y$$.

We need to find what values of $$x$$ and $$y$$ solve this equation. There are three possible cases.
1. $$x$$ and $$y$$ are real, unique and not equal.
2. $$x$$ and $$y$$ are real and equal.
3. $$x$$ and $$y$$ are complex.

The key to determining which case holds is to look at the determinant of the coefficient matrix, i.e.
$$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$
If this determinant is zero, then we cannot use this technique and either case 2 or 3 hold. If this case is non-zero, then case 1 holds and we can solve this problem. Let's call the coefficient matrix $$A$$ and so
$$\abs{A} = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$$

Solution

As long as the determinant of the coefficient matrix is NOT zero, we can solve this system of equations and the values are given by
$$\displaystyle{ x = \frac{ \begin{vmatrix} z_0 & b \\ z_1 & d \end{vmatrix} }{\abs{A}} }$$       and       $$\displaystyle{ y = \frac{ \begin{vmatrix} a & z_0 \\ c & z_1 \end{vmatrix} }{\abs{A}} }$$

In each case above, notice that we have replaced the column of matrix $$A$$ corresponding to the variable we are calculating with the $$z$$ constants.

Here are a couple of videos with examples.

### PatrickJMT - video 1

video by PatrickJMT

### PatrickJMT - video 2

video by PatrickJMT

The best way to learn Cramer's Rule is by watching someone work specific problems and then working plenty on your own. To get started, here is a video showing, in general, how to use Cramer's Rule, then he does a specific example. After this video, you should be able to work problems on your own.

### Thinkwell - Using Cramer's Rule [7min-33secs]

video by Thinkwell

Here is a video showing a proof or justification for Cramer's Rule.

### PatrickJMT - Cramer's Rule : A Proof / Justification for a System of 2 Linear Equations, 2 Unknowns [10min-12secs]

video by PatrickJMT

Important!

Okay, now you are ready for some practice problems.

Practice

Solve these linear systems using Cramer's Rule.

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x+4y=-14;$$ $$-2x-3y=11$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x+4y=-14;$$ $$-2x-3y=11$$

$$x=2, y=-5$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x+4y=-14;$$ $$-2x-3y=11$$

Solution

### 1884 video

video by Krista King Math

$$x=2, y=-5$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$5x+7y=-1;$$ $$6x+8y=1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$5x+7y=-1;$$ $$6x+8y=1$$

$$x=15/2, y=11/2$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$5x+7y=-1;$$ $$6x+8y=1$$

Solution

### 1885 video

video by Thinkwell

$$x=15/2, y=11/2$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$4x-2y=10;$$ $$3x-5y=11$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$4x-2y=10;$$ $$3x-5y=11$$

$$x=2, y=-1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$4x-2y=10;$$ $$3x-5y=11$$

Solution

### 1894 video

video by mattemath

$$x=2, y=-1$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x-y=-3;$$ $$x+4y=17$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x-y=-3;$$ $$x+4y=17$$

$$x=1, y=4$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x-y=-3;$$ $$x+4y=17$$

Solution

### 1895 video

video by MIP4U

$$x=1, y=4$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x-3y=16;$$ $$x+2y=1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x-3y=16;$$ $$x+2y=1$$

$$x=5, y=-2$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x-3y=16;$$ $$x+2y=1$$

Solution

### 1897 video

video by MIP4U

$$x=5, y=-2$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 5y = 26$$
$$5x - 4y = -1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 5y = 26$$
$$5x - 4y = -1$$

$$(3, 4)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 5y = 26$$
$$5x - 4y = -1$$

Solution

### 2855 video

$$(3, 4)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x - 2y = -4$$
$$4x - y = 3$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x - 2y = -4$$
$$4x - y = 3$$

$$(2, 5)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x - 2y = -4$$
$$4x - y = 3$$

Solution

### 2856 video

$$(2, 5)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y = 13$$
$$3x - 5y = -9$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y = 13$$
$$3x - 5y = -9$$

$$(2, 3)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y = 13$$
$$3x - 5y = -9$$

Solution

### 2857 video

$$(2, 3)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x - 7y = 1$$
$$3x + y = 13$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x - 7y = 1$$
$$3x + y = 13$$

$$(4, 1)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x - 7y = 1$$
$$3x + y = 13$$

Solution

### 2858 video

$$(4, 1)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - 2y = -3$$
$$3x + y = 5$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - 2y = -3$$
$$3x + y = 5$$

$$(1, 2)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - 2y = -3$$
$$3x + y = 5$$

Solution

### 2859 video

$$(1, 2)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - y = 4$$
$$2x + y = 2$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - y = 4$$
$$2x + y = 2$$

$$(2, -2)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - y = 4$$
$$2x + y = 2$$

Solution

### 2860 video

$$(2, -2)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$4x - 2y = 10$$
$$3x - 5y = 11$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$4x - 2y = 10$$
$$3x - 5y = 11$$

$$(2, -1)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$4x - 2y = 10$$
$$3x - 5y = 11$$

Solution

### 2861 video

video by mattemath

$$(2, -1)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x - 3y = 1$$
$$x + 2y = 11$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x - 3y = 1$$
$$x + 2y = 11$$

$$(5, 3)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x - 3y = 1$$
$$x + 2y = 11$$

Solution

### 2862 video

$$(5, 3)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x = 2y + 5$$
$$4y = 6x - 8$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x = 2y + 5$$
$$4y = 6x - 8$$

no solution

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x = 2y + 5$$
$$4y = 6x - 8$$

Solution

### 2863 video

no solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x + 2y = -3$$
$$-3x - 6y = 9$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x + 2y = -3$$
$$-3x - 6y = 9$$

infinite number of solutions

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x + 2y = -3$$
$$-3x - 6y = 9$$

Solution

### 2864 video

infinite number of solutions

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x + 7y = 5$$
$$-2x + y = -9$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x + 7y = 5$$
$$-2x + y = -9$$

$$(4, -1)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x + 7y = 5$$
$$-2x + y = -9$$

Solution

### 2865 video

$$(4, -1)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y = -4$$
$$-x + y = 7$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y = -4$$
$$-x + y = 7$$

$$(-5, 2)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y = -4$$
$$-x + y = 7$$

Solution

### 2866 video

$$(-5, 2)$$

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Now you are ready for some 3x3 systems.

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x+3y+z=2;$$ $$-x+2y+3z=-1;$$ $$-3x-3y+z=0$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x+3y+z=2;$$ $$-x+2y+3z=-1;$$ $$-3x-3y+z=0$$

$$x=4, y=-3, z=3$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x+3y+z=2;$$ $$-x+2y+3z=-1;$$ $$-3x-3y+z=0$$

Solution

### 1896 video

video by MIP4U

$$x=4, y=-3, z=3$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x-y-z=2;$$ $$4x+y-z=-5;$$ $$6x-2y+2z=17$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x-y-z=2;$$ $$4x+y-z=-5;$$ $$6x-2y+2z=17$$

$$x=1/2, y=-4, z=3$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x-y-z=2;$$ $$4x+y-z=-5;$$ $$6x-2y+2z=17$$

Solution

### 1902 video

video by MIP4U

$$x=1/2, y=-4, z=3$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x+2z=9;$$ $$2y+z=8;$$ $$4x-3y=-2$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x+2z=9;$$ $$2y+z=8;$$ $$4x-3y=-2$$

$$x=1, y=2, z=4$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x+2z=9;$$ $$2y+z=8;$$ $$4x-3y=-2$$

Solution

### 1903 video

video by PatrickJMT

$$x=1, y=2, z=4$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x-y+z=4;$$ $$2x+y+z=7;$$ $$-x-2y+2z=-1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x-y+z=4;$$ $$2x+y+z=7;$$ $$-x-2y+2z=-1$$

$$x=3, y=0, z=1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x-y+z=4;$$ $$2x+y+z=7;$$ $$-x-2y+2z=-1$$

Solution

### 1904 video

video by PatrickJMT

$$x=3, y=0, z=1$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$-x+2y-3z=1;$$ $$2x+z=0;$$ $$3x-4y+4z=2$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$-x+2y-3z=1;$$ $$2x+z=0;$$ $$3x-4y+4z=2$$

$$x=-4/5, y=-3/2, z=-8/5$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$-x+2y-3z=1;$$ $$2x+z=0;$$ $$3x-4y+4z=2$$

Solution

### 1905 video

video by mattemath

$$x=-4/5, y=-3/2, z=-8/5$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - 3y + 3z = -6$$
$$-2x + 4y + z = 3$$
$$3x - 5y +4z = -9$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - 3y + 3z = -6$$
$$-2x + 4y + z = 3$$
$$3x - 5y +4z = -9$$

$$(0, 1, -1)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x - 3y + 3z = -6$$
$$-2x + 4y + z = 3$$
$$3x - 5y +4z = -9$$

Solution

This problem is solved in 2 consecutive videos.

### 2867 video

$$(0, 1, -1)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x - 2y + z = 2$$
$$4x + 3y - 2z = 4$$
$$5x - 3y + 3z = 8$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x - 2y + z = 2$$
$$4x + 3y - 2z = 4$$
$$5x - 3y + 3z = 8$$

$$(1, 2, 3)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x - 2y + z = 2$$
$$4x + 3y - 2z = 4$$
$$5x - 3y + 3z = 8$$

Solution

### 2868 video

$$(1, 2, 3)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x + y - z = -2$$
$$2x - y + z = 0$$
$$x - 2y + 3z = 1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x + y - z = -2$$
$$2x - y + z = 0$$
$$x - 2y + 3z = 1$$

$$(-2/3, -7/3, -1)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$x + y - z = -2$$
$$2x - y + z = 0$$
$$x - 2y + 3z = 1$$

Solution

### 2870 video

$$(-2/3, -7/3, -1)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + y - z = 1$$
$$3x + 2y + 2z = 13$$
$$4x - 2y + 3z = 9$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + y - z = 1$$
$$3x + 2y + 2z = 13$$
$$4x - 2y + 3z = 9$$

$$(1, 2, 3)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + y - z = 1$$
$$3x + 2y + 2z = 13$$
$$4x - 2y + 3z = 9$$

Solution

### 2871 video

$$(1, 2, 3)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y - 5z = 1$$
$$x + y - z = 2$$
$$2y + z = 8$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y - 5z = 1$$
$$x + y - z = 2$$
$$2y + z = 8$$

$$(1, 3, 2)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$2x + 3y - 5z = 1$$
$$x + y - z = 2$$
$$2y + z = 8$$

Solution

### 2872 video

$$(1, 3, 2)$$

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x + y - z = 0$$
$$-2x + 5y + 4z = -1$$
$$3x + 2y + z = 1$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x + y - z = 0$$
$$-2x + 5y + 4z = -1$$
$$3x + 2y + z = 1$$

$$(1/2, -2/3, 5/6)$$

Problem Statement

Unless otherwise instructed, solve this linear system using Cramer's Rule.
$$3x + y - z = 0$$
$$-2x + 5y + 4z = -1$$
$$3x + 2y + z = 1$$

Solution

### 2873 video

$$(1/2, -2/3, 5/6)$$

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Really UNDERSTAND Precalculus

 basics of matrices matrix determinants

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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