# Trigonometry Problems set6

In this page trigonometry problems set6 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 20

Simplify the following

Solution:

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

In the denominator we have (1 + sin θ) and (1 - sin θ) and it looks like the algebraic formula  (a+b) (a-b) . So we have used the formula and got 1² - sin² θ. In the numerator multiplying (1 + sin θ) by cos θ and (1 - sin θ) by cos θ we get

negative cos θ sin θ and positive cos θ sin θ will get canceled.

identity for 1² - sin² θ is cos² θ

Question 21

Simplify the following

Solution:

L.H.S

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

In the denominator we have (cosec θ + 1) and (cosec θ - 1) and it looks like the algebraic formula  (a+b) (a-b) . So we have used the formula and got cosec² θ - 1. In the numerator multiplying (cosec θ - 1) by cos θ and (cosec θ + 1) by cos θ we get

negative cos θ and positive cos θ will get canceled.

identity for cosec² θ - 1  is cot² θ.

cot² θ can be written as cos² θ/sin² θ

we have two fractions in both numerator and denominator .We have written the fraction in the numerator as it is and the fraction which is in denominator can be written as its reciprocal.

Question 22

Simplify the following

Solution:

= (1 - cos θ)/(1 + cos θ)

In the first step we need to multiply both numerator and denominator by the conjugate of the denominator of the given fraction. The denominator of the given fraction is (1 + cos θ) then its conjugate must be 1 - cos θ.

we can write 1 - cos² θ as sin² θ. Now we can take only one square instead of putting squares for both numerator and denominator. So that we will get

= (Cosec θ - cot θ)²

Trigonometry Problems Set6 to Trigonometry