MHT CET · Maths · Trigonometric Ratios & Identities
The value of \(\cos 20^{\circ}+2 \sin ^2 55^{\circ}-\sqrt{2} \sin 65^{\circ}\) is
- A 0
- B 1
- C -1
- D \(\frac{1}{2}\)
Answer & Solution
Correct Answer
(B) 1
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \cos 20^{\circ}+2 \sin ^2 55^{\circ}-\sqrt{2} \sin 65^{\circ} \\ & =\cos 20^{\circ}+1-\cos 2\left(55^{\circ}\right)-\sqrt{2} \sin 65^{\circ} \\ & \ldots\left[2 \sin ^2 \theta=1-\cos 2 \theta\right] \\ & =\cos 20^{\circ}-\cos 110^{\circ}-\sqrt{2} \sin 65^{\circ}+1 \\ & =2 \sin 65^{\circ} \sin 45^{\circ}-\sqrt{2} \sin 65^{\circ}+1 \\ & =2 \sin 65^{\circ}\left(\frac{1}{\sqrt{2}}\right)-\sqrt{2} \sin 65^{\circ}+1 \\ & =\sqrt{2} \sin 65^{\circ}-\sqrt{2} \sin 65^{\circ}+1 \\ & =1\end{aligned}\)
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