Question

(sec^4 theta -sec^2 theta)=tan^2 theta+ tan^4 theta​

Answer 1

To prove ---> Sec⁴θ - Sec²θ = tan²θ + tan⁴θ

Proof ---> LHS = Sec⁴θ - Sec²θ

Taking Sec²θ common , we get

= Sec²θ ( Sec²θ - 1 )

We have an identity , Sec²θ - 1 = tan²θ , applying it here we get

= Sec²θ ( tan²θ )

= tan²θ Sec²θ

We have an identity , Sec²θ = 1 + tan²θ , applying it here , we get

= tan²θ ( 1 + tan²θ )

= tan²θ + tan⁴θ = RHS

Additional identities ---->

(1) Sin²θ + Cos²θ = 1

Sin²θ = 1 - Cos²θ

Cos²θ = 1 - Sin²θ

(2) 1 + Cot²θ = Cosec²θ

Cosec²θ - Cot²θ = 1

Cot²θ = Cosec²θ - 1