Prove the following:2.sec2 θ – sec4 θ – 2.cosec2 θ + cosec4 θ

1.	Prove the following:2.sec2 θ – sec4 θ – 2.cosec2 θ + cosec4 θ = cot4 θ – tan4 θ
Answer» LHS = 2.sec² θ – sec⁴ θ – 2.cosec² θ + cosec⁴ θ = 2 sec² θ – (sec² θ)² – 2cosec² θ + (cosec² θ)² = 2(1+ tan² θ) – (1+ tan² θ)² – 2(1+ cot² θ) + (1+ cot² θ)² = 2 + 2tan² θ – (1 + 2tan² θ + tan⁴ θ) – 2 – 2cot² θ + 1 + 2cot² θ + cot⁴ θ = 2 + 2.tan² θ – 1 – 2 tan² θ – tan⁴ θ – 2 – 2 cot² θ + 1 + 2 cot² θ + cot⁴ θ = cot⁴ θ – tan⁴ θ = R.H.S.

Prove the following:2.sec2 θ – sec4 θ – 2.cosec2 θ + cosec4 θ = cot4 θ – tan4 θ

Answer»

LHS = 2.sec² θ – sec⁴ θ – 2.cosec² θ + cosec⁴ θ = 2 sec² θ – (sec² θ)² – 2cosec² θ + (cosec² θ)²

= 2(1+ tan² θ) – (1+ tan² θ)² – 2(1+ cot² θ) + (1+ cot² θ)²

= 2 + 2tan² θ – (1 + 2tan² θ + tan⁴ θ) – 2 – 2cot² θ + 1 + 2cot² θ + cot⁴ θ

= 2 + 2.tan² θ – 1 – 2 tan² θ – tan⁴ θ – 2 – 2 cot² θ + 1 + 2 cot² θ + cot⁴ θ

= cot⁴ θ – tan⁴ θ = R.H.S.