Question : If $\cot A=\frac{5}{12}$, find the value of the following expression:
$\frac{5\left(1-\cos^2 A\right)}{6\left(1-\sin^2 A\right)}$
Option 1: $\frac{144}{5}$
Option 2: $\frac{24}{25}$
Option 3: $\frac{144}{25}$
Option 4: $\frac{24}{5}$
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Correct Answer: $\frac{24}{5}$
Solution :
Given: $\cot A=\frac{5}{12}⇒ \cot^2A = \frac{25}{144}$
Using identity: $\cos^2A+\sin^2A=1$
$\frac{5(1-\cos^2A)}{6(1-\sin^2A)}$ = $\frac{5\sin^2A}{6\cos^2A}$
= $\frac{5}6\times \tan^2A $
= $\frac{5}6\times\frac{1}{\cot^2A} $
= $\frac{5}6\times \frac{144}{ 25}$
= $\frac{24}5$
Hence, the correct answer is $\frac{24}{5}$.
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