Question

If $\alpha$ and $\beta$ are acute angles such that $co{s}^2\alpha +co{s}^2\beta =3/2$ and sin$\alpha$ . sin $\beta$ = 1/4 , then $\alpha +\beta$ equals

• A. ${30}^0$
• B. ${45}^0$
• C. ${60}^0$
• D. ${90}^0$

Answer 1

Answer: (C)

${60}^0$

${\cos }^2\alpha +{\cos }^2\beta =3/2$ and $sin\alpha sin\beta =\frac{1}{4}$

$\frac{1+\cos 2\alpha }{2}+\frac{1+\cos 2\beta }{2}=3/2$ and $2sin\alpha sin\beta =\frac{1}{2}$

$\cos 2\alpha +\cos 2\beta =1$ and $\cos (\alpha -\beta )-\cos (\alpha +\beta )=\frac{1}{2}$ ...... (ii)

$2\cos (\alpha +\beta )\cos (\alpha -\beta )=1$ ....... (i)

From (i) and (ii)

$\cos (\alpha +\beta )\cos (\alpha -\beta )=\cos (\alpha -\beta )-\cos (\alpha +\beta )$

$\Rightarrow \cos (\alpha +\beta )=\frac{1}{2}$ and $\cos (\alpha -\beta )=1$

$\alpha +\beta =\frac{\pi }{3}$