Question : Which of the following will satisfy $a^2 = b^2 + (ab)^2$ for the values $a$ and $b$?
Option 1: $a = \sin x \text{ and } b= \cot x$
Option 2: $a = \cos x \text{ and } b = \tan x$
Option 3: $a = \cot x \text{ and } b = \cos x$
Option 4: $a = \sin x \text{ and } b = \tan x$
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Correct Answer: $a = \cot x \text{ and } b = \cos x$
Solution :
Put $b=\cos x$ and $a=\cot x$ in $a^2 = b^2 + (ab)^2$ to verify the relation.
$\cos^2 x + ( \cot x \cos x)^2$
$= \cos^2 x ( 1+ \cot^2 x)$
$= \cos^2 x × \operatorname{cosec}^2 x $
$= \cos^2 x × \frac{1}{\sin^2 x} $
$= \cot^2 x $
So, $\cot^2x=\cos^2 x + ( \cot x \cos x)^2$
⇒ The values of $a$ and $b$ are $\cot x$ and $\cos x$ respectively.
Hence, the correct answer is $a = \cot x \text{ and } b = \cos x$.
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