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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Trignometry
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If $\tan\theta=\large\frac{b}{a}$ then find the value of $a^2\cos 2\theta+b\sin 2\theta$

$(a)\;a\qquad(b)\;a^2\qquad(c)\;a^3\qquad(d)\;b^2$

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1 Answer

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$a^2\cos 2\theta+b\sin 2\theta$
$\Rightarrow a^2\large\frac{1-\tan^2\theta}{1+\tan^2\theta}+$$b\large\frac{2\tan\theta}{1+\tan^2\theta}$
$\Rightarrow \large\frac{a^2\big[1-\large\frac{b^2}{a^2}\big]}{1+\Large\frac{b^2}{a^2}}+\frac{b.2b/a}{1+\Large\frac{b^2}{a^2}}$
$\Rightarrow \large\frac{a^2\big(\Large\frac{a^2-b^2}{a^2}\big)}{\Large\frac{a^2+b^2}{a^2}}+\frac{\Large\frac{2b^2}{a}}{\Large\frac{a^2+b^2}{b^2}}$
$\Rightarrow \large\frac{a^2(a^2-b^2)}{a^2+b^2}+\frac{2b^2.a}{a^2+b^2}$
$\Rightarrow \large\frac{a^2(a^2-b^2+2b^2)}{a^2+b^2}$
$\Rightarrow \large\frac{a^2(a^2+b^2)}{a^2+b^2}$
$\Rightarrow a^2$
Hence (b) is the correct answer.
answered Oct 16, 2013 by sreemathi.v
 

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