# Trigonometry Problems set3

In this page trigonometry problems set3 we are going to see practice questions in this topic.Here we you can find solution with detailed explanation.

## Identities of Trigonometry

Let us see trigonometric-identities

1. sin² θ  + cos² θ = 1
2. sin² θ  = 1 - cos² θ
3. cos² θ = 1 - sin² θ
4. Sec² θ - tan² θ = 1
5. Sec² θ  = 1 +  tan² θ
6. tan² θ  =  Sec² θ - 1
7. Cosec² θ - cot² θ = 1
8. Cosec² θ = 1 + cot² θ
9. cot² θ =  Cosec² θ - 1

These identities are applied in both ways ,left to right or right to left.So we have to memories all the identities.

Question 10

Prove that (sec θ + cos θ) (sec θ - cos θ) = tan²θ + sin²θ

Solution:

L.H.S

= (sec θ + cos θ) (sec θ - cos θ)

Let us compare this terms with the following algebraic identity.Here instead of a we have sec θ and instead of b we have cos θ.

= (sec² θ - cos² θ)

 trigonometry problems set3

From this identity we can derive formula for sec² θ that is 1 + tan² θ and we can derive for cos² θ that is 1 - sin² θ.

= [(1 + tan² θ) - (1 - sin² θ)]

distributing negative for 1 - sin² θ we are getting

= [1 + tan² θ - 1 + sin² θ]

= tan² θ + sin² θ

R.H.S

Question 11

Simplify the following

Solution:

Since the denominators are not same we need to take L.C.M for two fractions.

removing bracket we get

negative sin θ and positive sin θ will get cancelled.

1 - sin² θ will become cos² θ.

Question 12

Simplify the following

Solution:

Since the denominators are not same we need to take L.C.M for two fractions.

applying the formula (a+b)² to expand (1+sin θ)². Here instead of a we have 1 and instead of b we have sin θ.

now apply the value for sin²θ + cos²θ that is 1.

Now we can take 2 from both terms in the numerator. Then we will get

both (1+sinθ) which is in numerator and denominator will get cancelled.

Question 13

Prove the following

Solution:

we can use 1- cos²θ  instead of sin²θ.

comparing 1 - cos²θ with algebraic formula a² - b² = (a + b)(a - b) we can write (1 + cosθ) (1 + cosθ)

both (1 + cosθ) which are in the numerator and denominator will get cancelled.

Trigonometry Problems Set3 to Trigonometry