Question

How do you prove that: `(sin^2 x+ cos^2 x + cot^2 x) / (1+tan^2 x) = cot^2 x` ?

Answer 1

We wish to prove that:

`(sin^2 x+ cos^2 x + cot^2 x) / (1+tan^2 x) -= cot^2 x`

We can utilise the identities:

`sin^2A + cos^2A -= 1`
`1 + tan^2A -= sec^2 A`
`1 + cot^2A -= csc^2 A`

Consider the LHS:

`LHS = (sin^2 x+ cos^2 x + cot^2 x) / (1+tan^2 x)`

`\ \ \ \ \ \ \ \ = (1 + cot^2 x) / (1+tan^2 x)`

`\ \ \ \ \ \ \ \ = (csc^2 x) / (sec^2 x)`

`\ \ \ \ \ \ \ \ = ((csc x) / (sec x))^2`

`\ \ \ \ \ \ \ \ = ((1/sinx) / (1/cosx))^2`

`\ \ \ \ \ \ \ \ = (cosx/sinx )^2`

`\ \ \ \ \ \ \ \ = (cotx)^2`

`\ \ \ \ \ \ \ \ = cot^2x`

`\ \ \ \ \ \ \ \ = RHS \ \ \ \ ` QED