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यदि $\cos \,(\theta - \alpha ) = a,\,\,\sin \,(\theta - \beta ) = b,\,\,$ हो, तब ${\cos ^2}(\alpha - \beta ) + 2ab\,\sin \,(\alpha - \beta )$ बराबर है
$4{a^2}{b^2}$
${a^2} - {b^2}$
${a^2} + {b^2}$
$ - {a^2}{b^2}$
Solution
(c) यहाँ $\sin (\alpha – \beta ) = \sin (\theta – \beta – \overline {\theta – \alpha } )$
$ = \sin (\theta – \beta )\cos (\theta – \alpha ) – \cos (\theta – \beta )\sin (\theta – \alpha )$
$ = ba – \sqrt {1 – {b^2}} \sqrt {1 – {a^2}} $
एवं $\cos (\alpha – \beta ) = \cos (\theta – \beta – \overline {\theta – \alpha } )$
$ = \cos (\theta – \beta )\cos (\theta – \alpha ) + \sin (\theta – \beta )\sin (\theta – \alpha )$
$ = a\sqrt {1 – {b^2}} + b\sqrt {1 – {a^2}} $
$\therefore $ अतः दिया गया व्यंजक ${\cos ^2}(\alpha – \beta ) + 2ab\sin (\alpha – \beta )$
$ = (a\sqrt {1 – {b^2}} + b\sqrt {1 – {a^2}{)^2}} + 2ab\{ ab – \sqrt {1 – {a^2}} \sqrt {1 – {b^2}} \} $
$ = {a^2} + {b^2}$.
ट्रिक : $\alpha = 30^\circ ,\beta = 60^\circ ,$$\theta = 90^\circ $ रखने पर
$a = \frac{1}{2},b = \frac{1}{2}$
$\therefore {\cos ^2}(\alpha – \beta ) + 2ab\sin (\alpha – \beta ) = \frac{3}{4} + \frac{1}{2} \times \left( { – \frac{1}{2}} \right) = \frac{1}{2}$
जो कि विकल्प $(c)$ द्वारा दिया जाता है।