Question

Prove that : Cos 2A =cot^2-1/cot^2+1​

Answer 1

`\cos(2A) = { \cos }^(2) A - { \sin }^(2) A \\ = { \cos }^(2) A - \frac{1}{ \csc {}^(2) A} \\ \\ = \frac{( { \cos}^(2) A. \csc {}^(2)A ) - 1}{ { \csc }^(2) A} \\ \\ = \frac{( \frac{ { \cos}^(2)A }{ { \sin }^(2) A}) - 1 }{ { \csc }^(2) A} \\ \\ = \frac{ { \cot}^(2)A - 1 }{ { \csc}^(2) A}`

but csc²A = cot²A + 1:

`= \frac{ { \cot}^(2)A - 1 }{ { \cot }^(2)A + 1 }`

# proved