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

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