Precalculus by Richard Wright

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Whoever loves money never has enough; whoever loves wealth is never satisfied with their income. This too is meaningless. Ecclesiastes‬ ‭5‬:‭10‬ ‭NIV‬‬

# 5-03 Verify Trigonometric Identities

Summary: In this section, you will:

• Verify trigonometric identities algebraically.
• Verify trigonometric identities graphically.

SDA NAD Content Standards (2018): PC.5.1

The length of a shadow can be calculated from a complex trigonometric function based on the angle of elevation of the sun. Trigonometric identities can be used to simplify the function.

## Verify Trigonometric Identities

To verify trigonometric identities, use the fundamental identities to simplify one side of the identity to make it look like the other side of the equation.

###### Things to Try for Verifying Identities
• Work with one side of the equation at a time. Usually start with the more complicated side.
• Try factoring or adding fractions.
• Look for places to use fundamental trigonometric identities.
• Try converting everything to sine and cosine.
• Try something. Even failure teaches you something.

#### Example 1: Verify a Trigonometric Identity

Verify $$\left(1 - \tan α \right)\left(1 + \tan α \right) = 2 - \sec^2 α$$.

###### Solution

$$\left(1 - \tan α \right)\left(1 + \tan α \right) = 2 - \sec^2 α$$

Multiply

$$\textcolor{blue}{1 - \tan^2 α} = 2 - \sec^2 α$$

Pythagorean identity ($$\tan^2 u + 1 = \sec^2 u$$) solved for tangent $$\tan^2 u = \sec^2 u - 1$$.

$$1 - \left(\textcolor{red}{\sec^2 α - 1}\right) = 2 - \sec^2 α$$

Simplify

$$\textcolor{green}{2 - \sec^2 α} = 2 - \sec^2 α$$

##### Note:

Graph both sides of the identity on the same coordinate plane. The graphs will be identical if it is an identity.

#### Example 2: Verify a Trigonometric Identity

Verify $$\sin^2 x - \sin^4 x = \cos^2 x - \cos^4 x$$

###### Solution

Both sides are equally complicated, so either side will be a good starting point. Let’s start with the right side.

$$\sin^2 x - \sin^4 x = \cos^2 x - \cos^4 x$$

Factor out cos2 x

$$\sin^2 x - \sin^4 x = \textcolor{blue}{\cos^2 x}\left(1 - \cos^2 x\right)$$

Use the Pythagorean identity ($$\sin^2 u + \cos^2 u = 1$$) solved for cos2 u ($$\cos^2 u = 1 - \sin^2 u$$).

$$\sin^2 x - \sin^4 x = \left(\textcolor{red}{1 - \sin^2 x}\right)\left(1 - \left(\textcolor{red}{1 - \sin^2 x}\right)\right)$$

Simplify

$$\sin^2 x - \sin^4 x = \left(1 - \sin^2 x\right)\left(\textcolor{green}{\sin^2 x}\right)$$

Multiply

$$\sin^2 x - \sin^4 x = \sin^2 x - \sin^4 x$$

##### Try It 1

Verify $$\tan^4 x - \sec^4 x = -1 - 2\tan^2 x$$.

#### Example 3: Verify a Trigonometric Identity

Verify $$\frac{\tan^2 x}{\sec x} = \sec x - \cos x$$.

###### Solution

Left side will be easier to work with.

$$\frac{\tan^2 x}{\sec x} = \sec x - \cos x$$

Use the Pythagorean identity ($$1 + \tan^2 u = \sec^2 u$$) solved for tan2 u ($$\tan^2 u = \sec^2 u - 1$$).

$$\frac{\textcolor{blue}{\sec^2 x - 1}}{\sec x} = \sec x - \cos x$$

Separate the fraction into two fractions.

$$\frac{\textcolor{red}{\sec^2 x}}{\sec x} - \frac{\textcolor{red}{1}}{\sec x} = \sec x - \cos x$$

Simplify the fractions and $$\frac{1}{\sec x} = \cos x$$.

$$\sec x - \textcolor{green}{\cos x} = \sec x - \cos x$$

#### Example 4: Verify a Trigonometric Identity

Verify $$\frac{1}{\csc x\cot x} = \sec x - \cos x$$.

###### Solution

The left is slightly more complicated, so start there.

$$\frac{1}{\csc x\cot x} = \sec x - \cos x$$

Since the fraction is $$\frac{1}{something}$$ it looks like a reciprocal. Use the reciprocal identities.

$$\textcolor{blue}{\sin x \tan x} = \sec x - \cos x$$

An identity that relates tangent and sine is $$\tan u = \frac{\sin u}{\cos u}$$.

$$\sin x \textcolor{red}{\frac{\sin x}{\cos x}} = \sec x - \cos x$$

$$\frac{\textcolor{red}{\sin^2 x}}{\cos x} = \sec x - \cos x$$

Use the Pythagorean identity $$\sin^2 u + \cos^2 u = 1$$ to substitute for sin2 x.

$$\frac{\textcolor{green}{1 - \cos^2 x}}{\cos x} = \sec x - \cos x$$

Separate the fraction into two fractions.

$$\frac{\textcolor{brown}{1}}{\cos x} - \frac{\textcolor{brown}{\cos^2 x}}{\cos x} = \sec x - \cos x$$

Simplify the fractions and $$\frac{1}{\cos x} = \sec x$$.

$$\textcolor{purple}{\sec x} - \cos x = \sec x - \cos x$$

##### Try It 2

Verify $$\frac{1}{\cos x \tan x} = \csc x$$

#### Example 5: Verify a Trigonometric Identity

Verify $$\sin\left(-θ \right)\sec\left(-θ \right) = -\tan θ$$.

###### Solution

The left side will be easier to work with.

$$\sin\left(-θ \right)\sec\left(-θ \right) = -\tan θ$$

Use the even/odd identities ($$\sin\left(-u\right) = -\sin u$$ and $$\sec\left(-u\right) = \sec u$$).

$$\textcolor{blue}{-\sin θ \sec θ} = -\tan θ$$

Use the reciprocal identity ($$\sec u = \frac{1}{\cos u}$$) to rewrite sec θ.

$$-\sin θ \textcolor{red}{\frac{1}{\cos θ}} = -\tan θ$$

$$-\frac{\sin θ}{\cos θ} = -\tan θ$$

Use the quotient identity ($$\tan u = \frac{\sin u}{\cos u}$$).

$$\textcolor{green}{-\tan θ} = -\tan θ$$

#### Example 6: Verify a Trigonometric Identity

Verify $$\frac{\sin x + \tan y}{\sin x \tan y} = \cot y + \csc x$$.

###### Solution

Again the left side is more complex.

$$\frac{\sin x + \tan y}{\sin x \tan y} = \cot y + \csc x$$

Separate the fraction into two fractions.

$$\frac{\textcolor{blue}{\sin x}}{\sin x \tan y} + \frac{\textcolor{blue}{\tan y}}{\sin x \tan y} = \cot y + \csc x$$

Simplify.

$$\textcolor{red}{\frac{1}{\tan y}} + \textcolor{red}{\frac{1}{\sin x}} = \cot y + \csc x$$

Use reciprocal identities ($$\cot u = \frac{1}{\tan u}$$ and $$\csc u = \frac{1}{\sin u}$$).

$$\textcolor{green}{\cot y} + \textcolor{green}{\csc x} = \cot y + \csc x$$

##### Try It 3

Verify $$\tan\left(\frac{π}{2} - x\right)\sin \left(-x\right) = -\cos x$$.

#### Example 7: Verify a Trigonometric Identity

Verify $$1 - \sin^2\left(\frac{π}{2} - x\right) = \sin^2 x$$ a) algebraically and b) graphically.

###### Solution
1. The left side is again the most complex.

$$1 - \sin^2\left(\frac{π}{2} - x\right) = \sin^2 x$$

Use the cofunction identity ($$\sin\left(\frac{π}{2} - u\right) = \cos u$$).

$$1 - \textcolor{blue}{\cos^2 x} = \sin^2 x$$

Use the Pythagorean identity ($$\sin^2 u + \cos^2 u = 1$$) solve for sin2 u ($$\sin^2 u = 1 - \cos^2 u$$).

$$\textcolor{green}{\sin^2 x} = \sin^2 x$$

2. Graph the left side of the equation, then graph the right side of the equation and notice that they are the same.

##### Try It 4

Verify $$\frac{\sec x - \csc x}{\sec x \csc x} = \sin x - \cos x$$ both (a) algebraically and (b) graphically.

##### Lesson Summary

###### Things to Try for Verifying Identities
• Work with one side of the equation at a time. Usually start with the more complicated side.
• Try factoring or adding fractions.
• Look for places to use fundamental trigonometric identities.
• Try converting everything to sine and cosine.
• Try something. Even failure teaches you something.

## Practice Exercises

1. Derive the other two Pythagorean identities from sin2 u + cos2 u = 1.
2. Verify the identities.

3. $$\left(1 - \cos t\right)\left(1 + \cos t\right) = \sin^2 t$$
4. $$\left(\csc α +1\right)\left(\csc α - 1\right) = \cot^2 α$$
5. $$\cos x - \cos^3 x = \cos x \sin^2 x$$
6. $$\sin^4 x - \cos^4 x = 1 - 2\cos^2 x$$
7. $$\frac{\sin x - 1}{\cos x} = \tan x - \sec x$$
8. $$\frac{\csc^2 x}{\tan x} = \cot x + \cot^3 x$$
9. $$\frac{1}{\cos x \sin x} = \cot x + \tan x$$
10. $$\tan\left(-t\right)\cos\left(-t\right)\cot\left(-t\right) = \cos t$$
11. $$\sin\left(-x\right)\csc\left(\frac{π}{2} - x\right) = -\tan x$$
12. $$\frac{\cos x + \cot y}{\cos x \cot y} = \sec x + \tan y$$
13. $$\frac{1}{1 - \sin x} + \frac{1}{1 + \sin x} = 2\sec^2 x$$
14. Verify the identity algebraically and graphically.

15. $$\tan\left(\frac{π}{2} - x\right)\tan x = 1$$
16. $$\sin^4 x = \sin^2 x - \sin^2 x \cos^2 x$$
17. The length, ℓ, of a shadow cast by a vertical stick of height, h, when the angle of elevation of the sun is θ can be modeled by

$$ℓ = \frac{h \cos θ}{\cos\left(90° - θ\right)}$$

Verify that $$ℓ = h \cot θ$$.

18. Mixed Review

19. (5-02) Factor and then use the fundamental trigonometric identities to simplify: $$\sin^4 x - 2\sin^2 x + 1$$.
20. (5-02) Rewrite the fraction so it is not in fractional form: $$\frac{3}{\csc x \tan x}$$.
21. (5-01) Simplify: $$\sin x\left(\csc x - \sin x\right)$$.
22. (4-02) List all the angles, θ, on the unit circle where $$\sin θ = ±\frac{\sqrt{3}}{2}$$.
23. (3-03) Condense: log 2x + 3 log x − 4 log y.

1. a) $$\sin^2 u + \cos^2 u = 1$$ → $$\frac{\sin^2 u + \cos^2 u}{\sin^2 u} = \frac{1}{\sin^2 u}$$ → $$1 + \cot^2 u = \csc^2 u$$; b) $$\sin^2 u + \cos^2 u = 1$$ → $$\frac{\sin^2 u + \cos^2 u}{\cos^2 u} = \frac{1}{\cos^2 u}$$ → $$\tan^2 u + 1 = \sec^2 u$$
2. $$\left(1 - \cos t\right)\left(1 + \cos t\right) =$$ $$1 - \cos^2 t =$$ $$\sin^2 t$$
3. $$\left(\csc α +1\right)\left(\csc α - 1\right) =$$ $$\csc^2 α - 1 =$$ $$\left(\cot^2 α + 1\right) - 1 =$$ $$\cot^2 α$$
4. $$\cos x - \cos^3 x =$$ $$\cos x\left(1 - \cos^2 x\right) =$$ $$\cos x \sin^2 x$$
5. $$\sin^4 x - \cos^4 x =$$ $$\left(\sin^2 x - \cos^2 x\right)\left(\sin^2 x + \cos^2 x\right) =$$ $$\left(\sin^2 x - \cos^2 x\right)\left(1\right) =$$ $$\sin^2 x - \cos^2 x =$$ $$1 - \cos^2 x - \cos^2 x =$$ $$1 - 2\cos^2 x$$
6. $$\frac{\sin x - 1}{\cos x} =$$ $$\frac{\sin x}{\cos x} - \frac{1}{\cos x} =$$ $$\tan x - \sec x$$
7. $$\frac{\csc^2 x}{\tan x} =$$ $$\frac{1 + \cot^2 x}{\tan x} =$$ $$\frac{1}{\tan x} + \frac{\cot^2 x}{\tan x} =$$ $$\cot x + \cot^3 x$$
8. $$\frac{1}{\cos x \sin x} =$$ $$\frac{\cos x}{\cos^2 x \sin x} =$$ $$\sec^2 x \cot x =$$ $$\frac{\sec^2 x}{\tan x} =$$ $$\frac{1 + \tan^2 x}{\tan x} =$$ $$\frac{1}{\tan x} + \frac{\tan^2 x}{\tan x} =$$ $$\cot x + \tan x$$
9. $$\tan\left(-t\right)\cos\left(-t\right)\cot\left(-t\right) =$$ $$\tan\left(-t\right)\cot\left(-t\right)\cos\left(-t\right) =$$ $$\cos\left(-t\right) =$$ $$\cos t$$
10. $$\sin\left(-x\right)\csc\left(\frac{π}{2} - x\right) =$$ $$-\sin x \sec x =$$ $$-\frac{\sin x}{\cos x} =$$ $$-\tan x$$
11. $$\frac{\cos x + \cot y}{\cos x \cot y} =$$ $$\frac{\cos x}{\cos x \cot y} + \frac{\cot y}{\cos x \cot y} =$$ $$\frac{1}{\cot y} + \frac{1}{\cos x} =$$ $$\sec x + \tan y$$
12. $$\frac{1}{1 - \sin x} + \frac{1}{1 + \sin x} =$$ $$\frac{1 + \sin x}{\left(1 - \sin x\right)\left(1 + \sin x\right)} + \frac{1 - \sin x}{\left(1 - \sin x\right)\left(1 + \sin x\right)} =$$ $$\frac{2}{1 - \sin^2 x} =$$ $$\frac{2}{\cos^2 x} =$$ $$2\sec^2 x$$
13. $$\tan\left(\frac{π}{2} - x\right)\tan x =$$ $$\cot x \tan x =$$ $$1$$
14. $$\sin^4 x =$$ $$\sin^2 x \sin^2 x =$$ $$\sin^2 x\left(1 - \cos^2 x\right) =$$ $$\sin^2 x - \sin^2 x \cos^2 x$$
15. $$ℓ = \frac{h \cos θ}{\cos\left(90° - θ\right)} =$$ $$\frac{h \cos θ}{\sin θ} =$$ $$h \cot θ$$
16. cos4 x
17. 3 cos x
18. cos2 x
19. $$\frac{π}{3}, \frac{2π}{3}, \frac{4π}{3}, \frac{5π}{3}$$
20. $$\log \frac{2x^4}{y^4}$$