If =a b , prove that : b 2 +a 2 =b - Vidyadip

Question

If $\tan \theta=\frac{a}{b}$, prove that :
$
b \cos 2 \theta+a \sin 2 \theta=b
$

✓

Answer

Given, $\tan \theta=\frac{a}{b}$
Now, L.H.S. $=b \cos 2 \theta+a \sin 2 \theta$
Using the formula
$=b \cdot \frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}+a \cdot \frac{2 \tan \theta}{1+\tan ^2 \theta}$
Putting the value
$=b \cdot \frac{1-\left(\frac{a}{b}\right)^2}{1+\left(\frac{a}{b}\right)^2}+2 a \cdot \frac{\frac{a}{b}}{1+\left(\frac{a}{b}\right)^2}$
$=b \cdot \frac{b^2-a^2}{b^2+a^2}+\frac{2 a^2 b}{b^2+a^2}$
$=\frac{b\left(b^2-a^2+2 a^2\right)}{\left(b^2+a^2\right)}=\frac{b\left(b^2+a^2\right)}{\left(b^2+a^2\right)}=b=$ R.H.S.

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