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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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Verifying a Trigonometric Identity

Verify that sin2xtan2x=1sin2x .

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 .

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

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

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 , and isolated the cos2x=1sin2x .

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 , we could have also substituted the identity cot2x=csc2x1 .

Example B

Verify the identity sinx1cosx=1+cosxsinx .

Solution: Multiply the left-hand side of the equation by 1+cosx1+cosx .

\frac{\sin x}{1- \cos x}&= \frac{1+ \cos x}{\sin x} \\
\frac{1+ \cos x}{1+ \cos x} \cdot \frac{\sin x}{1- \cos x}&= \\
\frac{\sin \left(1+\cos x\right)}{1- \cos^2x}&= \\
\frac{\sin \left(1+\cos x\right)}{\sin^2x}&= \\
\frac{1+\cos x}{\sin x}&=

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

Example C

Verify the identity sec(x)=secx .

Solution: Change secant to cosine.

sec(x)=1cos(x)

From the Negative Angle Identities, we know that cos(x)=cosx .

&=\frac{1}{\cos x} \\
&=\sec x

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 , we get cos2x=1sin2x .

cos2x=cos2x and the equation is verified.

Guided Practice

Verify the following identities.

1. cosxsecx=1

2. 2sec2x=1tan2x

3. cos(x)1+sin(x)=secx+tanx

Answers

1. Change secant to cosine.

\cos x \sec x&=\cos \cdot \frac{1}{\cos x} \\
&=1

2. Use the identity 1+tan2θ=sec2θ .

2- \sec^2x&=2-(1+ \tan^2x) \\
&=2-1- \tan^2x \\
&=1- \tan^2x

3. Here, start with the Negative Angle Identities and multiply the top and bottom by 1+sinx1+sinx to make the denominator a monomial.

\frac{\cos \left(-x\right)}{1+ \sin \left(-x\right)}&=\frac{\cos x}{1- \sin x} \cdot \frac{1+ \sin x}{1+ \sin x} \\
&=\frac{\cos x \left(1+ \sin x \right)}{1- \sin^2x} \\
&=\frac{\cos x \left(1+ \sin x\right)}{\cos^2x} \\
&=\frac{1+ \sin x}{\cos x} \\
&=\frac{1}{\cos x}+ \frac{\sin x}{\cos x} \\
&=\sec x+ \tan x

Explore More

Verify the following identities.

  1. cot(x)=cotx
  2. csc(x)=cscx
  3. tanxcscxcosx=1
  4. sinx+cosxcotx=cscx
  5. csc(π2x)=secx
  6. tan(π2x)=tanx
  7. cscxsinxcotxtanx=1
  8. tan2xtan2x+1=sin2x
  9. (sinxcosx)2+(sinx+cosx)2=2
  10. sinxsinxcos2x=sin3x
  11. tan2x+1+tanxsecx=1+sinxcos2x
  12. cos2x=cscxcosxtanx+cotx
  13. 11sinx11+sinx=2tanxsecx
  14. csc4xcot4x=csc2x+cot2x
  15. (sinxtanx)(cosxcotx)=(sinx1)(cosx1)

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