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If $\cot A+\frac{1}{\cot A}=2$, find the value of $\left(\cot ^{2} A+\frac{1}{\cot ^{2} A}\right)$



Given : $\cot A+\frac{1}{\cot A}=2$

$\cot A+\frac{1}{\cot A}=2$

Squaring both sides, we get

$\Rightarrow\left(\cot A+\frac{1}{\cot A}\right)^{2}=2^{2}$

$\Rightarrow \cot ^{2} A+\left(\frac{1}{\cot A}\right)^{2}+2(\cot A)\left(\frac{1}{\cot A}\right)=4$

$\Rightarrow \cot ^{2} A+\frac{1}{\cot ^{2} A}+2=4$

$\Rightarrow \cot ^{2} A+\frac{1}{\cot ^{2} A}=4-2$

$\Rightarrow \cot ^{2} A+\frac{1}{\cot ^{2} A}=2$

Hence, the value of $\left(\cot ^{2} A+\frac{1}{\cot ^{2} A}\right)$ is 2 .


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