If $\tan A = \frac{1}{2}$ and $\tan B = \frac{1}{3}$,then $\cos 2A = $

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

Prove that: $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$

If $\tan A = \frac{1}{\sqrt{x(x^2+x+1)}}, \tan B = \frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C = (x^{-3}+x^{-2}+x^{-1})^{\frac{1}{2}}$,where $0 < A, B, C < \frac{\pi}{2}$,then $A+B$ is equal to:

Evaluate: $\cos^2 48^\circ - \sin^2 12^\circ$

If $\alpha$ is in the $3^{\text{rd}}$ quadrant,$\beta$ is in the $2^{\text{nd}}$ quadrant such that $\tan \alpha = \frac{1}{7}$ and $\sin \beta = \frac{1}{\sqrt{10}}$,then find the value of $\sin(2\alpha + \beta)$.