Prove that

Question

csc x - cot x = tan (x/2)? 

 

Answer 1

The trigonometric equation is csc x - cot x = tan(x/2).

Left hand side identity : csc x - cot x.

Reciprocal identity : csc x = 1/sin x and cot x = 1/tan x.

= 1/sin x - 1/tan x

Half - angle formulas : sin x = 2 tan (x/2) / (1 + tan² (x/2)),

                                   tan x = 2 tan (x/2) / (1 - tan² (x/2)).

= [(1 + tan² (x/2)) / 2 tan (x/2)] - [(1 - tan² (x/2)) / 2 tan (x/2)]

= [1 + tan² (x/2) - 1 + tan² (x/2)] / 2 tan (x/2)

= 2tan² (x/2) / 2 tan (x/2)

= tan (x/2)

= Right hand side identity.

Hence proved.