Question

Prove the following:

cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)

Answer 1

We have to prove that,

cot4x (sin5x + sin3x) = cot x (sin5x – sin3x)

i.e., to prove that,

`(sin5x + sin3x)/(sin5x - sin3x) = cotx/(cot4x)`

L.H.S. = `(sin5x + sin3x)/(sin5x - sin3x)`

= `(2sin((5x + 3x)/2)*cos((5x - 3x)/2))/(2cos((5x + 3x)/2)*sin((5x - 3x)/2)`

= `(2sin4x*cosx)/(2cos4x*sinx)`

= tan4x · cotx

= `cotx/(cot4x)`

= R.H.S.