How to prove this trigonometric identity?

Question

Autumn Armstrong · Accepted Answer

See below

#cos2x=cos^2x-sin^2# #sin2x=2sinxcosx# Are you happy about these identities? If so then proving identities, as above, is a matter of doing something and seeing where you get to!!! LHS can be written #(cosx-(cos^2x-sin^2x)+2)/(3sinx-2sinxcosx)# =#(cosx-cos^2x+sin^2x+2)/(sinx(3-2cosx))# RHS #(1+cosx)/sinx# Multiply LHS and RHS by #sinx(3-2cosx)# LHS=#cosx-cos^2x+sin^2x+2# RHS=#(1+cosx)(3-2cosx)# =#3-2cosx+3cosx-2cos^2x# =#3+cosx-2cos^2x# Now looking at the LHS and remembering that #sin^2x+cos^2x=1#or # sin^2x=1-cos^2x# LHS=#cosx-cos^2x+(1-cos^2x)+2# =#cosx-2cos^2x+3# We are there!!! Difficult for me to write this really clearly here, paper would be easier!!! BUT THE IMPORTANT THING WITH IDENTITIES IS DO SOMETHING AND YOU WILL GET THERE.

Mason Adams · Answer

undefined To prove a trigonometric identity, you typically use algebraic manipulation and properties of trigonometric functions. Here are the general steps:

1. Start with one side of the identity and manipulate it using known trigonometric identities, algebraic properties, and trigonometric identities to transform it into the other side.
2. Use identities such as Pythagorean identities, sum and difference identities, double angle identities, and other trigonometric identities to simplify the expression.
3. Rearrange terms and simplify until you reach the other side of the identity.

It's important to have a good understanding of trigonometric identities and properties to effectively prove them. Practice and familiarity with common identities are also helpful.