What is the value of sin2 θ+11+tan2 θ?

Question:

What is the value of $\sin ^{2} \theta+\frac{1}{1+\tan ^{2} \theta} ?$

Solution:

We have,

$\sin ^{2} \theta+\frac{1}{1+\tan ^{2} \theta}=\sin ^{2} \theta+\frac{1}{\sec ^{2} \theta}$

$=\sin ^{2} \theta+\left(\frac{1}{\sec \theta}\right)^{2}$

$=\sin ^{2} \theta+\cos ^{2} \theta$

$=1$

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