^2 ( 4 - ) - ^2 ( - 4 ) = - Vidyadip

MCQ

${\cos ^2}\left( {\frac{\pi }{4} - \beta } \right) - {\sin ^2}\left( {\alpha - \frac{\pi }{4}} \right) = $

A
$\sin (\alpha + \beta )\sin (\alpha - \beta )$
B
$\cos (\alpha + \beta )\cos (\alpha - \beta )$
C
$\sin (\alpha - \beta )\cos (\alpha + \beta )$
✓
$\sin (\alpha + \beta )\cos (\alpha - \beta )$

✓

Answer

Correct option: D.

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

d
(d) ${\cos ^2}\left( {\frac{\pi }{4} - \beta } \right) - {\sin ^2}\left( {\alpha - \frac{\pi }{4}} \right)$

$ = \cos \,\left( {\frac{\pi }{4} - \beta + \alpha - \frac{\pi }{4}} \right)\,\cos \,\left( {\frac{\pi }{4} - \beta - \alpha + \frac{\pi }{4}} \right)\,$

$ = \cos (\alpha - \beta )\cos \left( {\frac{\pi }{2} - \overline {\alpha + \beta } } \right) $

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

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