If cot θ=13√, find the value of 1−cos2 θ2−sin2 θ.

Question:

If $\cot \theta=\frac{1}{\sqrt{3}}$, find the value of $\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}$.

Solution:

Given: $\cot \theta=\frac{1}{\sqrt{3}}$

We have to find the value of the expression $\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}$

We know that,

$1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$

$\Rightarrow \operatorname{cosec}^{2} \theta=1+\left(\frac{1}{\sqrt{3}}\right)^{2}$

$\Rightarrow \operatorname{cosec}^{2} \theta=\frac{4}{3}$

Using the identity $\sin ^{2} \theta+\cos ^{2} \theta=1$, we have

$\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}=\frac{\sin ^{2} \theta}{2-\sin ^{2} \theta}$

$=\frac{\frac{1}{\operatorname{cosec}^{2} \theta}}{2-\frac{1}{\operatorname{cosec}^{2} \theta}}$

$=\frac{1}{2 \operatorname{cosec}^{2} \theta-1}$

$=\frac{1}{2 \times \frac{4}{3}-1}$

$=\frac{3}{5}$

Hence, the value of the given expression is $\frac{3}{5}$.

 

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