Solving Trigonometric Equations

The various trigonometric formulae and identities can be used to help solve trigonometric equations. Here is a summary of the most important trigonometric formulae you should know:
sin²x + cos²x = 1
1 + cot²x = cosec²x
tan²x + 1 = sec²x
cos2x = cos²x - sin²x = 2cos²x – 1 = 1 - 2sin²x
sin2x = 2sinx cosx
tanx = sinx
          cosx
The compound angle formulae: remember also the useful technique of writing expressions in the form rcos(q + a)

See also: solving basic equations

Proving Identities

You may be asked to use the formulae above to prove new trigonometric identities. To prove an identity, start from one side and manipulate it until you get the other side.

Example

Prove that cosx cos2x + sinx sin2x = cosx

Starting from the left hand side:
cosx cos2x + sinx sin2x = cos(x - 2x)
= cos(-x) = cosx