Q63 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics
$\cos ^2 76^{\circ}+\sin ^2 46^{\circ}+\sin 76^{\circ} \cos 46^{\circ}=$
- A$\frac{3}{4}$
- B1
- Correct answerC$\frac{5}{4}$
- D$2$
Solution
We have
$\begin{aligned} & \cos ^2 76^{\circ}+\sin ^2 46^{\circ}+\sin 76^{\circ} \cos 46^{\circ} \\ & \Rightarrow \cos ^2\left(30^{\circ}+46^{\circ}\right)+\sin ^2 46^{\circ}+\sin \left(30^{\circ}+46^{\circ}\right) \cos 46^{\circ} \\ & \Rightarrow\left(\cos 30^{\circ} \cos 46^{\circ}-\sin 30^{\circ} \sin 46^{\circ}\right)^2+\sin ^2 46^{\circ}+\left(\sin 30^{\circ}\right. \\ & \left.\cos 46^{\circ}+\cos 30^{\circ} \sin 46^{\circ}\right) \cos 46^{\circ} \\ & =3 \frac{\cos ^2 46^{\circ}}{4}+\frac{1}{4} \sin ^2 46^{\circ}-2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} \cos 46^{\circ} \cdot \sin 46^{\circ} \\ & +\sin ^2 46+\frac{1}{2} \cos ^2 46^{\circ}+\frac{\sqrt{3}}{2} \sin 46^{\circ} \cos 46^{\circ} \\ & =\frac{5}{4}\left(\sin ^2 46^{\circ}+\cos ^2 46^{\circ}\right)=\frac{5}{4} \\ & \end{aligned}$
Hence, the correct answer is option: 'C'.