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# Proofs of Trigonometric Identities

## Convert to sine/cosine, use basic identities, and simplify sides of the equation.

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Practice Proofs of Trigonometric Identities
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Verifying a Trigonometric Identity

Verify that sin2xtan2x=1sin2x\begin{align*}\frac{\sin^2x}{\tan^2x}=1 - \sin^2x\end{align*} .

### Guidance

This concept continues where the previous one left off. Now that you are comfortable simplifying expressions, we will extend the idea to verifying entire identities. Here are a few helpful hints to verify an identity:

• Change everything into terms of sine and cosine.
• Use the identities when you can.
• Start with simplifying the left-hand side of the equation, then, once you get stuck, simplify the right-hand side. As long as the two sides end up with the same final expression, the identity is true.

#### Example A

Verify the identity cot2xcscx=cscxsinx\begin{align*}\frac{\cot^2x}{\csc x}=\csc x - \sin x\end{align*} .

Solution: Rather than have an equal sign between the two sides of the equation, we will draw a vertical line so that it is easier to see what we do to each side of the equation. Start with changing everything into sine and cosine.

cot2xcscxcos2xsin2x1sinxcos2xsinxcscxsinx1sinxsinx\begin{align*}\begin{array}{c|c c} \frac{\cot^2x}{\csc x} & \csc x - \sin x \\ \frac{\frac{\cos^2x}{\sin^2x}}{\frac{1}{\sin x}} & \frac{1}{\sin x}- \sin x \\ \frac{\cos^2x}{\sin x}\end{array}\end{align*}

Now, it looks like we are at an impasse with the left-hand side. Let’s combine the right-hand side by giving them same denominator.

1sinxsin2xsinx1sin2xsinxcos2xsinx\begin{align*}\begin{array}{|c} \frac{1}{\sin x}- \frac{\sin^2x}{\sin x} \\ \frac{1- \sin^2x}{\sin x} \\ \frac{\cos^2x}{\sin x}\end{array}\end{align*}

The two sides reduce to the same expression, so we can conclude this is a valid identity. In the last step, we used the Pythagorean Identity, sin2θ+cos2θ=1\begin{align*}\sin^2 \theta+\cos^2 \theta=1\end{align*} , and isolated the cos2x=1sin2x\begin{align*}\cos^2x=1- \sin^2x\end{align*} .

There are usually more than one way to verify a trig identity. When proving this identity in the first step, rather than changing the cotangent to cos2xsin2x\begin{align*}\frac{\cos^2x}{\sin^2x}\end{align*} , we could have also substituted the identity cot2x=csc2x1\begin{align*}\cot^2x=\csc^2x-1\end{align*} .

#### Example B

Verify the identity sinx1cosx=1+cosxsinx\begin{align*}\frac{\sin x}{1- \cos x}=\frac{1+ \cos x}{\sin x}\end{align*} .

Solution: Multiply the left-hand side of the equation by 1+cosx1+cosx\begin{align*}\frac{1+ \cos x}{1+ \cos x}\end{align*} .

sinx1cosx1+cosx1+cosxsinx1cosxsin(1+cosx)1cos2xsin(1+cosx)sin2x1+cosxsinx=1+cosxsinx====

The two sides are the same, so we are done.

#### Example C

Verify the identity sec(x)=secx\begin{align*}\sec(-x)=\sec x\end{align*} .

Solution: Change secant to cosine.

sec(x)=1cos(x)

From the Negative Angle Identities, we know that cos(x)=cosx\begin{align*}\cos (-x)=\cos x\end{align*} .

=1cosx=secx

Concept Problem Revisit Start by simplifying the left-hand side of the equation.

sin2xtan2x=sin2xsin2xcos2x=cos2x
.

Now simplify the right-hand side of the equation. By manipulating the Trigonometric Identity,

sin2x+cos2x=1\begin{align*}\sin^2x + \cos^2x = 1\end{align*} , we get cos2x=1sin2x\begin{align*}\cos^2x = 1 - \sin^2x\end{align*} .

cos2x=cos2x\begin{align*}\cos^2x =\cos^2x\end{align*} and the equation is verified.

### Guided Practice

Verify the following identities.

1. cosxsecx=1\begin{align*}\cos x \sec x=1\end{align*}

2. 2sec2x=1tan2x\begin{align*}2- \sec^2x=1- \tan^2x\end{align*}

3. cos(x)1+sin(x)=secx+tanx\begin{align*}\frac{\cos \left(-x\right)}{1+ \sin \left(-x\right)}=\sec x+ \tan x\end{align*}

1. Change secant to cosine.

cosxsecx=cos1cosx=1

2. Use the identity 1+tan2θ=sec2θ\begin{align*}1+ \tan^2 \theta=\sec^2 \theta\end{align*} .

2sec2x=2(1+tan2x)=21tan2x=1tan2x

3. Here, start with the Negative Angle Identities and multiply the top and bottom by 1+sinx1+sinx\begin{align*}\frac{1+ \sin x}{1+ \sin x}\end{align*} to make the denominator a monomial.

cos(x)1+sin(x)=cosx1sinx1+sinx1+sinx=cosx(1+sinx)1sin2x=cosx(1+sinx)cos2x=1+sinxcosx=1cosx+sinxcosx=secx+tanx

### Explore More

Verify the following identities.

1. cot(x)=cotx\begin{align*}\cot (-x)=- \cot x\end{align*}
2. csc(x)=cscx\begin{align*}\csc (-x)=- \csc x\end{align*}
3. tanxcscxcosx=1\begin{align*}\tan x \csc x \cos x=1\end{align*}
4. sinx+cosxcotx=cscx\begin{align*}\sin x+ \cos x \cot x=\csc x\end{align*}
5. csc(π2x)=secx\begin{align*}\csc \left(\frac{\pi}{2}-x\right)=\sec x\end{align*}
6. tan(π2x)=tanx\begin{align*}\tan \left(\frac{\pi}{2}-x\right)=\tan x\end{align*}
7. cscxsinxcotxtanx=1\begin{align*}\frac{\csc x}{\sin x}- \frac{\cot x}{\tan x}=1\end{align*}
8. tan2xtan2x+1=sin2x\begin{align*}\frac{\tan^2x}{\tan^2x+1}=\sin^2x\end{align*}
9. (sinxcosx)2+(sinx+cosx)2=2\begin{align*}(\sin x- \cos x)^2+(\sin x+ \cos x)^2=2\end{align*}
10. sinxsinxcos2x=sin3x\begin{align*}\sin x- \sin x \cos^2x= \sin^3x\end{align*}
11. tan2x+1+tanxsecx=1+sinxcos2x\begin{align*}\tan^2x+1+\tan x \sec x=\frac{1+ \sin x}{\cos^2x}\end{align*}
12. cos2x=cscxcosxtanx+cotx\begin{align*}\cos^2x=\frac{\csc x \cos x}{\tan x+ \cot x}\end{align*}
13. 11sinx11+sinx=2tanxsecx\begin{align*}\frac{1}{1- \sin x} - \frac{1}{1+ \sin x}=2 \tan x \sec x\end{align*}
14. csc4xcot4x=csc2x+cot2x\begin{align*}\csc^4x- \cot^4x=\csc^2x+\cot^2x\end{align*}
15. (sinxtanx)(cosxcotx)=(sinx1)(cosx1)\begin{align*}(\sin x - \tan x)(\cos x- \cot x)=(\sin x-1)(\cos x-1)\end{align*}