Question

Csc(x) - cot(x) forms an identity with which of the following?

a. tan(x) b. tan (x/2) c. cot (x/2) d. cot (x)

Answer 1

csc ( x ) - cot ( x ) = ( 1 / sin ( x) ) - ( cos (x) / sin (x) )
= ( 1 - cos (x) ) / sin (x)

then 1 - cos (x) = 2 sin^2 ( x/2 )
and sin x = 2sin( x/2)cos( x/2)

substitute it
= ( 1 - cos (x) ) / sin (x)
=2 sin^2 ( x/2 ) / 2sin( x/2)cos( x/2)

then simplify
= sin (x/2) / cos (x/2)
since sin/cos = tan

csc ( x ) - cot ( x ) = tan ( x/2)