Given
[tex]\frac{\cos ^2\theta+\sin ^2\theta}{1+\tan ^2\theta}[/tex]
To simplify this expression, you have to use trigonometric identities.
The numerator of the expression corresponds to one of the Pythagorean identities which is:
[tex]\cos ^2\theta+\sin ^2\theta=1[/tex]
You can write the expression as follows:
[tex]\begin{gathered} \frac{\cos ^2\theta+\sin ^2\theta}{1+\tan ^2\theta} \\ \frac{1}{1+\tan ^2\theta} \end{gathered}[/tex]
The denominator of the expression corresponds to another trigonometric identity which is:
[tex]\sec ^2\theta=1+\tan ^2\theta[/tex]
Now you can write the expression as follows:
[tex]\begin{gathered} \frac{1}{1+\tan ^2\theta} \\ \frac{1}{\sec ^2\theta} \end{gathered}[/tex]
The inverse of the secant is equal to the cosine so that:
[tex]\frac{1}{\sec^2\theta}=\cos ^2\theta[/tex]
The answer is
[tex]\cos ^2\theta[/tex]
