(a) If a cos θ+ b sin θ= m and a sin θ- b cos θ= n, then prove that a^2+ b^2= m^2+ n^2.

If $\displaystyle \mathrm{a} \cos \theta+\mathrm{b} \sin \theta=\mathrm{m}$ and $\displaystyle \mathrm{a} \sin \theta-\mathrm{b} \cos \theta=\mathrm{n}$, then prove that $\displaystyle \mathrm{a}^{2}+\mathrm{b}^{2}=\mathrm{m}^{2}+\mathrm{n}^{2}$.

Prove that : \[\sqrt{\frac{\sec \mathrm{A}-1}{\sec \mathrm{~A}+1}}+\sqrt{\frac{\sec \mathrm{A}+1}{\sec \mathrm{~A}-1}}=2 \operatorname{cosec} \mathrm{~A} \]