# Trigonometry Problems set4

In this page trigonometry problems set4 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 14

Prove that sin θ (1 + tan θ) + cos θ (1 + cot θ) = sec θ + cosec θ

Solution:

L.H.S

= sin θ (1 + tan θ) + cos θ (1 + cot θ)

To simplify this first we are  going to apply the formulas for tanθ and cotθ Instead of tan θ we can write sin θ/cos θ. Like that instead of cot θ we can write cos θ/sin θ.

= sin θ (1 + sin θ/cos θ) + cos θ (1 + cos θ/sin θ) Since the denominators are not same let us take L.C.M for both fractions. Now we are going to take (cos θ + sin θ) from the numerator. Now we are plugging the value of cos ² θ + sin ² θ that is 1. both cos θ and sin θ will get cancelled in the numerator and denominators from the first and second fraction respectively. We have used reciprocal formula ti get the value of 1/sin θ and 1/cos θ.

Question 15

Prove that (1 - sin² θ) sec² θ = 1

Solution:

L.H.S

= (1 - sin² θ) sec² θ we can write cos² θ instead of 1 - sin² θ form the identity cos² θ + sin² θ = 1.

= (cos² θ) sec² θ in the next step we are going to write 1/cos²θ instead of sec² θ using reciprocal formula.

= (cos² θ) (1/cos² θ)

= (cos² θ/cos² θ)

= 1

R.H.S

Both cos² θ both are in the numerator and denominator will get cancelled. Finally we get 1.

Question 16

Prove that (1 + tan² θ) cos² θ = 1

Solution:

L.H.S

= (1 + tan² θ) cos² θ From this identity we come to know the value of (1 + tan² θ) that is sec²θ.

= (sec² θ) cos² θ                   trigonometry problems set4 = (1/cos² θ) cos² θ

= (cos² θ/cos² θ)

= 1

R.H.S

Trigonometry Problems Set4 to Trigonometry