Question

If this is solved you will be awarded a high mark prove the following formula 1 + sec²=cot²​

Answer 1

This relationship is not true.

`1 + \sec^2(x) = 1 + (1 - \tan^2(x)) = 2 - \tan^2(x)`

and

`\cot^2(x) = \frac1{\tan^2(x)}`

Rewrite the equation in terms of only tangent.

`2 - \tan^2(x) = \frac1{\tan^2(x)} \iff \tan^4(x) - 2\tan^2(x) + 1 = 0`

Factorize the left side.

`\left(\tan^2(x) - 1\right)^2 = 0`

Solve for
`\tan(x)`.

`\tan^2(x) - 1 = 0`

`\tan^2(x) = 1`

`\tan(x) = \pm 1`

`x = \pm\tan^(-1)(1) + n\pi = \pm\frac\pi4 + n\pi`

where
`n` is an integer. That is, the equation only holds for certain values of
`x`, as opposed to all
`x`, so it's not an identity.