<s>[INST] Show that x is independent of this trig expression For some reason I cannot get the solution.
Show that $x$ is independent of: $\sin^2(x+y)+\sin^2(x+z)-2\cos(y-z)\sin(x+y)\sin(x+z)$
I have used all the identities but seem to be missing something.
 [/INST] $\begin{align}
& \sin^2(x+y)+\sin^2(x+z)-2\cos(y-z)\sin(x+y)\sin(x+z) \\
&=\dfrac{1}{2}(1-\cos(2x+2y)) + \dfrac{1}{2}(1-\cos(2x+2z)) - 2\cos(y-z)\sin(x+y)\sin(x+z)\\
&=1-\cos(2x+y+z)\cos(y-z) -  2\cos(y-z)\sin(x+y)\sin(x+z)\\
&=1-\cos(y-z)\left[\cos(2x+y+z) + 2\sin(x+y)\sin(x+z) \right]\\
&=1-\cos(y-z)\left[\cos(2x+y+z) + \cos(y-z) - \cos(2x+y+z)\right]\\
&=1-\cos^2(y-z)\\
\end{align}$
which is independent of $x.$
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