sin(theta)-sin(5theta)/cos(theta)+cos(5theta) = -tan(2theta)

Question

Verify the Identity

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Answer 1

You can use other "identities" to prove this. You can prove these other identities first if you like.

I prove them below.

sin(theta) - sin(5 theta) = sin(3 theta - 2 theta) - sin(3 theta + 2 theta) = sin 3 theta cos 2 theta - sin 2 theta cos 3 theta - sin 3 theta cos 2 theta - cos 3 theta sin 2 theta = - 2 sin 2 theta cos 3 theta

cos(theta) + cos(5 theta) = cos(3 theta - 2 theta) + cos(3 theta + 2 theta) = cos(3 theta)cos(2 theta) + sin(3 theta sin 2 theta) + cos(3 theta)cos(2 theta) - sin(3 theta sin 2 theta) = 2 cos(3 theta)cos(2 theta)

Thus,(sin(theta) - sin(5 theta))/(cos(theta)+cos(5 theta)) =

- 2 sin 2 theta cos 3 theta/(2 cos(3 theta)cos(2 theta)) =

-sin(2 theta)/cos(2 theta) = - tan(2 theta) =

-tan(2 theta)