Question

The value of $\frac{(1+{\tan }^2\theta )}{(1+{\cot }^2\theta )}$ is

• A. ${\tan }^2\theta$
• B. ${\cot }^2\theta$
• C. ${\sec }^2\theta$
• D. ${\text{cosec}}^2\theta$

Answer 1

Answer: (A)

${\tan }^2\theta$

The value of $\frac{(1+{\tan }^2\theta )}{(1+{\cot }^2\theta )}$ is
$=\frac{(1+\frac{{\sin }^2\theta }{{\cos }^2\theta })}{(1+\frac{{\cos }^2\theta }{{\sin }^2\theta })}$
$=\frac{\left(\frac{{\cos }^2\theta +{\sin }^2\theta }{{\cos }^2\theta }\right)}{\left(\frac{{\sin }^2\theta +{\cos }^2\theta }{{\sin }^2\theta }\right)}$
$=\frac{\left(\frac{1}{{\cos }^2\theta }\right)}{\left(\frac{1}{{\sin }^2\theta }\right)}$
$=\frac{{\sin }^2\theta }{{\cos }^2\theta }$
$={\tan }^2\theta$
Hence, option A is correct.