Prove that


Prove that

(i) $\cos (n+2) x \cos (n+1) x+\sin (n+2) x \sin (n+1) x=\cos x$

(ii) $\cos \left(\frac{\pi}{4}-\mathrm{x}\right) \cos \left(\frac{\pi}{4}-\mathrm{y}\right)-\sin \left(\frac{\pi}{4}-\mathrm{x}\right) \sin \left(\frac{\pi}{4}-\mathrm{y}\right)=\sin (\mathrm{x}+\mathrm{y})$



(i) $\cos (n+2) x \cdot \cos (n+1) x+\sin (n+2) x \cdot \sin (n+1) x$

$=\sin ((n+2) x+(n+1) x)($ using $\cos (A-B)=\cos A \cos B+\sin A \sin B)$

$=\cos (n x+2 x-(n x+x))$

$=\cos (n x+2 x-n x-x)$

$=\cos x$

(ii) $\cos \left(\frac{\pi}{4}-x\right) \cos \left(\frac{\pi}{4}-y\right)-\sin \left(\frac{\pi}{4}-x\right) \sin \left(\frac{\pi}{4}-y\right)$

$=\cos \left(\frac{\pi}{4}-x+\frac{\pi}{4}-y\right)($ using $\cos (A+B)=\cos A \cos B-\sin A \sin B)$

$=\cos \left(\frac{2 \pi}{4}-x-y\right)$

$=\cos \left(\frac{2 \pi}{4}-\mathrm{x}-\mathrm{y}\right)$

$=\cos \left(\frac{\pi}{2}-(x+y)\right)\left(u \sin g \cos \left(\frac{\pi}{2}-x\right)=\sin x\right)$

$=\sin (x+y)$

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