Question

Prove that: tan²(sec^–1 2) + cot² (cosec^–1 3) = 11.

Answer 1

To prove: tan² (sec^–1 2) + cot² (cosec^–1 3) = 11

Let sec^–1 2 = θ

⇒ sec θ = 2

∴ tan² θ = sec² θ – 1

= 2² – 1

= 3

Also, let cosec^–1 3 = α

⇒ cosec α = 3

∴ cot² α = cosec² α – 1

= 3² – 1

= 8

Now, L.H.S. = tan² (sec^–1 2) + cot² (cosec^–1 3)

= tan² θ + cot² α

= 3 + 8

= 11

= R.H.S.

Hence Proved.