Question

2Sec^2theta - sec^4theta - 2cosec^2theta + cosec^4theta = cot^4theta - tan^4theta

Answer 1

Answer:

Step-by-step explanation:

Consider LHS:

2sec2θ-sec4θ-2cosec2θ+cosec4θ = 2sec2θ – 2cosec2θ – sec4θ + cosec4θ

= 2(sec2θ – cosec2θ) – (sec4θ –  cosec4θ)

= 2(sec2θ – cosec2θ) – (sec2θ – cosec2θ) (sec2θ + cosec2θ)

= (sec2θ – cosec2θ)[2 – (sec2θ + cosec2θ)]

= (sec2θ – cosec2θ)[2 – (1 + tan2θ + 1 + cot2θ)]

= (1 + tan2θ – 1 – cot2θ)[2 – (2 + tan2θ + cot2θ)]

= (tan2θ – cot2θ) [2 – 2 – tan2θ – cot2θ)]

= (tan2θ – cot2θ) × – [tan2θ + cot2θ)]

= (cot2θ – tan2θ) × [cot2θ + tan2θ)]

= cot4θ – tan4θ

= RHS