Question

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)​

Answer 1

Explanation:

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)

Answer 2

Explanation:

Use the identity tan(x) = sin(x) / cos(x) in the left hand side of the given identity.

tan2(x) - sin2(x) = sin2(x) / cos2(x) - sin2(x)

= [ sin2(x) - cos2(x) sin2(x) ] / cos2(x)

= sin2(x) [ 1 - cos2(x) ] / cos2(x)

= sin2(x) sin2(x) / cos2(x)

= sin2(x) tan2(x) which is equal to the right hand side of the given identity.