﻿ Proving Trigonometric Identities Worksheet | Problems & Solutions

# Proving Trigonometric Identities Worksheet

Proving Trigonometric Identities Worksheet
• Page 1
1.
Simplify: sin3A + sin Acos2A
 a. tan A b. sin 2A c. sin A d. cos A

#### Solution:

sin3A + sin Acos2A

= sin A(sin2A + cos2A)
[Factor out sin A.]

= sin A · 1
[Substitute: sin2A + cos2A = 1.]

= sin A

2.
Factorize: cos2A - sin2A cos2A
 a. sin4A b. cos4A c. cos2A d. sin2A

#### Solution:

cos2A - sin2A cos2A

= cos2A (1 - sin2A)
[Factor out cos2A.]

= cos2A · cos2A
[Substitute:1 - sin2A = cos2A.]

= cos4A

3.
Simplify: sin B + $\frac{{\mathrm{cos}}^{2}B}{\mathrm{sin}B}$
 a. sec B b. sin B c. csc B d. cos B

#### Solution:

sin B + cos2 Bsin B

= sin2 B+cos2 Bsin B

= 1sin B
[Substitute: sin2 B + cos2 B = 1.]

= csc B

4.
Find the value of sin θ + cos θ cot θ - csc θ.
 a. 2sec $\theta$ b. 1 c. 2csc $\theta$

#### Solution:

sin θ + cos θ cot θ - csc θ

= sin θ + cos θ(cos θsin θ ) - csc θ
[Substitute: cot θ = cos θsin θ .]

= sin θ + cos2θsin θ - (1sin θ)
[Substitute: csc θ = 1sin θ.]

= sin2θ+cos2θ - 1sin θ = 1- 1sin θ = 0
[Substitute: sin2 θ + cos2 θ = 1.]

5.
Simplify: sec θ -
 a. cos $\theta$ b. sec $\theta$ c. sin $\theta$ d. csc $\theta$

#### Solution:

sec θ - tan2θsec θ = sec2θ-tan2θsec θ

= 1sec θ
[Substitute: sec2θ - tan2θ = 1.]

= cos θ

6.
Find the value of 1 + ${\mathrm{sec}}^{2}\theta \left(1-{\mathrm{sin}}^{2}\theta \right)$.
 a. 1 b. ${\mathrm{tan}}^{2}\theta$ c. 2

#### Solution:

1 + sec2θ(1-sin2θ)

= 1 + sec2θ · cos2θ
[Substitute: 1 - sin2θ = cos2θ.]

= 1 + 1cos2θ · cos2θ

= 1 + 1 = 2

7.
1 - () = ___________.

 a. ${\mathrm{sec}}^{2}A$ b. 1 - ${\mathrm{tan}}^{2}A$ c. 1

#### Solution:

1 - (sin A  sec Atan A)

= 1 - sin A  (1cos A)tan A
[Substitute: sec A = 1cos A.]

= 1 - (tan Atan A)
[Substitute sin Acos A = tan A.]

= 1 - 1 = 0

8.
Find the value of - - - .
 a. 2 b. 1 c. 2

#### Solution:

sec4 A - tan4 A - sec2 A - tan2 A

= (sec4 A - sec2 A) - (tan4 A + tan2 A)
[Group the similar functions.]

= sec2 A (sec2 A - 1) - tan2 A(tan2 A + 1)

= sec2 Atan2 A - sec2 Atan2 A = 0
[Use: sec2 A - 1 = tan2 A and tan2 A + 1 = sec2 A.]

9.
Simplify:
cos θ ·
 a. tan2$\theta$ b. 1 c. cot2$\theta$

#### Solution:

cos θ · csc θcot θ

= cosθ  1sin θcot θ
[Substitute csc θ = 1sin θ.]

= cot θcot θ = 1
[Substitute cos θsin θ = cot θ.]

10.
Simplify:
 a. sec B - tan B b. sin B - sec B c. tan B - 1 d. tan B - sec B

#### Solution:

sin B - 1cos B = sin Bcos B - 1cos B

= tan B - sec B
[Substitute: sin Bcos B = tan B and 1cosB = sec B.]