Question

Prove the following:

i. sec² θ + cosec² θ = sec² θ × cosec² θ 

ii. cot² θ – tan² θ = cosec² θ – sec² θ

Answer 1

i. L.H.S. = sec θ + cosec θ

∴ sec² θ + cosec² θ = sec² θ × cosec² θ

ii. L.H.S. = cot² θ – tan² θ 

= (cosec² θ – 1) – (sec² θ – 1) [∵ tan² θ = sec² θ – 1, cot² θ = cosec² θ – 1] 

= cosec² θ – 1 – sec² θ + 1 

cosec² θ – sec² θ 

= R.H.S. 

∴ cot² θ – tan² θ = cosec² θ – sec² θ