If a = b, then a 2 + b 2 = - Vidyadip

MCQ

If $a \tan \theta= b$, then $a \cos 2 \theta+ b \sin 2 \theta=$

✓
a
B
b
C
-a
D
-b

✓

Answer

Correct option: A.

a

(A)
$a \cos 2 \theta+b \sin 2 \theta$
$=a\left(\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\right)+b\left(\frac{2 \tan \theta}{1+\tan ^2 \theta}\right)$
$=a\left(\frac{1-\frac{b^2}{a^2}}{1+\frac{b^2}{a^2}}\right)+b\left(\frac{\frac{2 b}{a}}{1+\frac{b^2}{a^2}}\right) \quad \ldots\left[\because \tan \theta=\frac{b}{a}(\right.$ given $\left.)\right]$
$=a\left(\frac{a^2-b^2}{a^2+b^2}\right)+b\left(\frac{2 b a}{a^2+b^2}\right)$
$=\frac{1}{\left(a^2+b^2\right)}\left(a^3-a b^2+2 a b^2\right)=\frac{a\left(a^2+b^2\right)}{a^2+b^2}$
= a

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