Prove the indentity 1 cosx cos2x sinx sin2x cotxSolutio

Prove the indentity

(1 + cos(x) + cos(2x)) / (sin(x) + sin(2x)) = cot(x)

Solution

Use the identities cos(2x) = 2 cos²(x) - 1 and sin(2x) = 2 sin(x) cos(x) in the left hand side of the given identity.

[ 1 + cos(x) + cos(2x) ] / [ sin(x) + sin(2x) ]

= [ 1 + cos(x) + 2 cos²(x) - 1 ] / [ sin(x) + 2 sin(x) cos(x) ]

= [ cos(x) + 2 cos²(x) ] / [ sin(x) + 2 sin(x) cos(x) ]

= cos(x) [1 + 2 cos(x)] / [ sin(x)( 1 + 2 cos(x) ) ]

= cot(x)