Prove trigonometric identity and determine a counterexample

[euro94]

Homework Statement

cos(x-y)cosy-sin(x-y)siny=cosx
a.try to prove that the equation is an identity
b. determine a counterexample to show that it is not an identity

Homework Equations

cos(x-y) = cosxcosy+sinxsiny
sin(x-y) = sinxcosy-cosxsiny

The Attempt at a Solution

a.Left side of equatioin: (cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
= cosxcosycos²y+sinxcosysinycosy - (sinxcosysiny - cosxsin²y)
I'm not sure where to go from there ...
b. how would i go about finding a counterexample?

 

[scurty]

(cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
= cosxcosycos2y +sinxcosysinycosy - (sinxcosysiny - cosxsin2y)

That part is wrong. Once you get that part right, consider trying this out: Factor cos(x) from two of the clusters of terms above and a simplification will happen.For the counterexample, just find a value of x and a valuye for y so that the equality doesn't hold.

 

[Curious3141]

Homework Statement

cos(x-y)cosy-sin(x-y)siny=cosx
a.try to prove that the equation is an identity
b. determine a counterexample to show that it is not an identity

Homework Equations

cos(x-y) = cosxcosy+sinxsiny
sin(x-y) = sinxcosy-cosxsiny

The Attempt at a Solution

a.Left side of equatioin: (cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
= cosxcosycos²y+sinxcosysinycosy - (sinxcosysiny - cosxsin²y)
I'm not sure where to go from there ...
b. how would i go about finding a counterexample?

Parts a and b are mutually exclusive. Either the relation given is an identity, or it is not. If it's an identity, you're supposed to prove it as in part a (in which case you don't have to answer part b). If it's not an identity, you can just provide a single counterexample for part b (in this case, you can't answer part a).

For this question, it is, in fact an identity. So only part a has an answer.

You know that cos(A+B) = cosAcosB - sinAsinB.

Now try letting A = x-y and B = y. What happens?