If = a b then (a -b ) (a +b ) =? - Vidyadip

MCQ

If $\tan\theta=\frac{\text{a}}{\text{b}}$ then $\frac{(\text{a}\sin\theta-\text{b}\cos\theta)}{(\text{a}\sin\theta+\text{b}\cos\theta)} =?$

A
$\frac{(\text{a}^2+\text{b}^2)}{(\text{a}^2-\text{b}^2)}$
✓
$\frac{(\text{a}^2-\text{b}^2)}{(\text{a}^2+\text{b}^2)}$
C
$\frac{\text{a}^2}{(\text{a}^2+\text{b}^2)}$
D
$\frac{\text{b}^2}{(\text{a}^2+\text{b}^2)}$

✓

Answer

Correct option: B.

$\frac{(\text{a}^2-\text{b}^2)}{(\text{a}^2+\text{b}^2)}$

Given, $\tan\theta=\frac{\text{a}}{\text{b}}$
Now, $\frac{(\text{a}\sin\theta-\text{b}\cos\theta)}{(\text{a}\sin\theta+\text{b}\cos\theta)} $
$=\frac{\frac{\text{a}\sin\theta}{\cos\theta}-\frac{\text{b}\cos\theta}{\cos\theta}}{\frac{\text{a}\sin\theta}{\cos\theta}+\frac{\text{b}\cos\theta}{\cos\theta}}$
$=\frac{\text{a}\tan\theta-\text{b}}{\text{a}\tan\theta+\text{b}}$
$=\frac{\text{a}\times\frac{\text{a}}{\text{b}}-\text{b}}{\text{a}\times\frac{\text{a}}{\text{b}}+\text{b}}$
$=\frac{\frac{\text{a}^2-\text{b}^2}{\text{b}}}{\frac{\text{a}^2+\text{b}^2}{\text{b}}}$
$=\frac{\text{a}^2-\text{b}^2}{\text{a}^2+\text{b}^2}$

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