(A)OR(B)

Evaluate $\displaystyle 2 \sec ^{2} \theta+3 \operatorname{cosec}^{2} \theta-2 \sin \theta \cos \theta$ if $\displaystyle \theta=45^{\circ}$.

\[\begin{aligned} & 2 \sec ^{2} 45^{\circ}+3 \operatorname{cosec}^{2} 45^{\circ}-2 \sin 45^{\circ} \cos 45^{\circ} \\ & =2(\sqrt{2})^{2}+3(\sqrt{2})^{2}-2\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right) \\ & =4+6-1=9 \end{aligned} \]

\[\begin{aligned} & \sin \theta-\cos \theta=0 \Rightarrow \sin \theta=\cos \theta \Rightarrow \tan \theta=1 \\ & \Rightarrow \theta=45^{\circ} \\ & \sin ^{4} 45^{\circ}+\cos ^{4} 45^{\circ}=\left(\frac{1}{\sqrt{2}}\right)^{4}+\left(\frac{1}{\sqrt{2}}\right)^{4} \\ & =\frac{1}{4}+\frac{1}{4}=\frac{1}{2} \end{aligned} \]