Question

Please do no 7 for me. I’m practising for my exam and can’t find the solution. The correct answer will get 15 points. No spam

Answer 1

Explanation:

Prove that

`( \tan(x) )/( \sec(x) - 1 ) + ( \sin(x) )/(1 + \cos(x) ) = 2 \csc(x)`

Recall that ta x = sin x/ cos x and sec x = 1/cos x. So the 1st term on the LHS becomes

`( ( \sin(x) )/( \cos(x) ) )/( (1)/( \cos(x) ) - 1 ) + ( \sin(x) )/(1 + \cos(x) ) = 2 \csc(x)`

or with the cos x cancelling out, the equation becomes

`( \sin(x) )/(1 - \cos(x) ) + ( \sin(x) )/(1 + \cos(x) ) = 2 \csc(x)`

Combining the terms on the LHS,

`(2 \sin(x) )/((1 - \cos(x))(1 + \cos(x) ) ) = 2 \csc(x)`

`\frac{2 \sin(x) }{1 - {( \cos(x) )}^(2) } = \frac{2 \sin(x) }{ { (\sin(x)) }^(2) } \\ or \\ (2)/( \sin(x) ) = 2 \csc(x)`