(A) Find the value of x for which ( sin A + cosec A )^2+( cos A + sec A )^2=x+ tan^2 A + cot^2 A

\[\begin{aligned} & (\sin \mathrm{A}+\operatorname{cosec} \mathrm{A})^{2}+(\cos \mathrm{A}+\sec \mathrm{A})^{2}=x+\tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A} \\ & \Rightarrow \sin ^{2} \mathrm{~A}+\operatorname{cosec}^{2} \mathrm{~A}+2+\cos ^{2} \mathrm{~A}+\sec ^{2} \mathrm{~A}+2=x+\tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A} \\ & \Rightarrow 1+2+2+1+\cot ^{2} \mathrm{~A}+1+\tan ^{2} \mathrm{~A}=x+\tan ^{2} \mathrm{~A}+\cot ^{2} \mathrm{~A} \\ & \therefore x=7 \end{aligned} \]

\[\begin{aligned} & \frac{3 \sin 30^{\circ}-4 \sin ^{3} 30^{\circ}}{2 \sin ^{2} 50^{\circ}+2 \cos ^{2} 50^{\circ}} \\ & =\frac{3 \times \dfrac{1}{2}-4 \times \dfrac{1}{8}}{2\left(\sin ^{2} 50^{\circ}+\cos ^{2} 50^{\circ}\right)} \\ & =\frac{\dfrac{3}{2}-\dfrac{1}{2}}{2 \times 1} \\ & =\frac{1}{2} \end{aligned} \]