Question

Prove that : sec⁴ θ – sec²θ = tan⁴θ + tan²θ.

Answer 1

Trigonometry is the study of the relationship between the sides and angles of a triangle.

An equation involving trigonometry ratios of an angle is called is called a trigonometric  identity, if it is true for all values of the angles involved. For any acute angle θ, we have 3 identities.

i) sin² θ + cos² θ = 1 ,ii) 1 + tan² θ = sec² θ , iii) cot² θ +1 = cosec² θ.

SOLUTION:

Given:
sec⁴ θ – sec²θ = tan⁴θ + tan²θ

LHS = sec⁴ θ – sec²θ
LHS= sec² θ (sec² θ -1)
LHS= (1 + tan² θ) (1 + tan² θ -1)

[ sec² θ = 1 + tan² θ ]

LHS= (1 + tan² θ) ( tan² θ)
LHS= tan²θ + tan⁴θ

LHS = RHS

HOPE THIS WILL HELP YOU..

Answer 2

We know that,
1 + tan²theta = sec²theta....
So,
LHS = Sec⁴theta -sec²theta
= (1+tan²theta) ²-(1+tan²theta)
= 1 +2tan²theta + tan⁴theta - 1-tan²theta
= tan⁴theta +tan²theta = RHS (Ans)...