# Trig proof help!

1. Oct 29, 2014

### stfz

1. The problem statement, all variables and given/known data
Prove that: $sin x + sin 2x + cos x + cos 2x = 2\sqrt{2} cos (\frac{x}{2}) sin(\frac{3x}{2}+\frac{\pi}{4})$

2. Relevant equations
We know all the double, compound, and half angle formulas.

3. The attempt at a solution
Taking on the RHS, we have

Expanding with half angle formula and compound angle formula

Hence we can cancel the sqrt(2) on the left, and replace sin/cos (pi/4) with exact values
And we have

If we expand the half angles in the right bracket

And then the sqrt(2) on the bottom can multiply to make 2, we can +/- them together

And cancel the twos.
Is it possible to cancel the +/- signs?
However, the graph of my result does not match that of the original.
For example, if I graph

As two graphs, one with a + and the other with a -, I find that the original graph
Is equal to the + graph in some intervals, and equal to the - graph in other intervals.

Thanks

Stephen

2. Oct 29, 2014

### PeroK

Have you ever seen this formula? $Asin(x) + Bcos(x) = \sqrt{A^2+B^2}sin(x+α) \ \ (α = tan^{-1}(\frac{B}{A}))$

3. Oct 29, 2014

### stfz

Hmm... Maybe that will work! Thanks for the suggestion, I will try it