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


Correct answer : (3)
 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


Correct answer : (2)
 3.  
Simplify: sin B + cos2BsinB
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


Correct answer : (3)
 4.  
Find the value of sin θ + cos θ cot θ - csc θ.
a.
2sec θ
b.
1
c.
2csc θ


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.]


Correct answer : (1)
 5.  
Simplify: sec θ - tan2θsec θ
a.
cos θ
b.
sec θ
c.
sin θ
d.
csc θ


Solution:

sec θ - tan2θsec θ = sec2θ-tan2θsec θ

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

= cos θ


Correct answer : (1)
 6.  
Find the value of 1 + sec2θ(1-sin2θ).
a.
1
b.
tan2θ
c.
2


Solution:

1 + sec2θ(1-sin2θ)

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

= 1 + 1cos2θ · cos2θ

= 1 + 1 = 2


Correct answer : (4)
 7.  
1 - (sin A  sec Atan A) = ___________.

a.
sec2A
b.
1 - tan2A
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


Correct answer : (4)
 8.  
Find the value of sec4 A - tan4 A - sec2 A - tan2 A.
a.
2
b.
1
c.
2tan2 Asec2 A


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.]


Correct answer : (1)
 9.  
Simplify:
cos θ · csc θcot θ
a.
tan2θ
b.
1
c.
cot2θ


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.]


Correct answer : (2)
 10.  
Simplify: sin B - 1cos B
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.]


Correct answer : (4)

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