(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. (b) Prove that: √((sec A -1)/(sec A +1))+√((sec A +1)/(sec A…

\[\begin{aligned} & m^{2}+n^{2}=(a \cos \theta+b \sin \theta)^{2}+(a \sin \theta-b \cos \theta)^{2} \\ & =a^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)+b^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right) \\ & =a^{2}+b^{2} \end{aligned} \]

LHS \(\displaystyle =\frac{\sqrt{\boldsymbol{\operatorname { s e c }} \mathbf{A}-\mathbf{1}}}{\sqrt{\boldsymbol{\operatorname { s e c }} \mathbf{~}} \mathbf{1}+\mathbf{1}}+\frac{\sqrt{\boldsymbol{s e c} \mathbf{A}+\mathbf{1}}}{\sqrt{\boldsymbol{s e c} \mathbf{A}-\mathbf{1}}}\)

\[\begin{aligned} & =\frac{\sec A-1+\sec A+1}{\sqrt{\sec ^{2} A-1}} \\ & =\frac{2 \sec A}{\tan A} \\ & =2 \operatorname{cosec} A=R H S \end{aligned} \]