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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Question : $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=$___________.
Question : Which of the following is equal to $\sec A – \cos A$?
Question : What will be the value of $\cos x \operatorname{cosec} x - \sin x \sec x$?
Question : The value of $\sec x - \cos x = $?
Question : Simplify: $\frac{\cos A}{1+\tan A}-\frac{\sin A}{1+\cot A}$
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