Question

Prove the identity 2csc2x=csc^2xtanz

Answer 1

Explanation:

Consider the LHS, after the 5th step, consider the RHS

`2 \csc(2x) = \csc {}^(2) (x) \tan(x)`

`2 (1)/( \sin(2x) ) = \csc {}^(2) (x) \tan(x)`

`2 * (1)/(2 \sin(x) \cos(x) ) = \csc {}^(2) (x) \tan(x)`

`(1)/( \sin(x) \cos(x) ) = \csc {}^(2) (x) \tan(x)`

`\csc(x) \sec(x) = \csc {}^(2) (x) \tan(x)`

Consider the RHS

`\csc(x) \sec(x) =( 1 + \cot {}^(2) (x) ( \tan(x))`

`\csc(x) \sec(x) = \tan(x) + \cot(x)`

`\csc(x) \sec(x) = ( \sin(x) )/( \cos(x) ) + ( \cos(x) )/( \sin(x) )`

`\csc(x) \sec(x) = (1)/( \sin(x) \cos(x) )`

`\csc(x) \sec(x) = \csc(x) \sec(x)`