Question

If sin x = p, then prove that `(1 + tan^2 x)/(1 + cot^2 x) = p^2/(1 - p^2)`.

Answer 1

To prove: `(1 + tan^2 x)/(1 + cot^2 x) = p^2/(1 - p^2)`

⇒ `(sec^2 x)/("cosec"^2 x)`   ...(By use of identity)

⇒ `(sin^2 x)/(cos^2 x)`

⇒ tan² x

⇒ `[p/sqrt(1 - p^2)]^2`   ...`[∵ tan x = 1/cot x]`

⇒ `p^2/(1 - p^2)`

∴ `(1 + tan^2 x)/(1 + cot^2 x) = p^2/(1 - p^2)`

Hence proved.