# How do you find the derivative of (cos^2(t) + 1)/(cos^2(t))?

Mar 30, 2016

$\frac{\text{d"}{"d} t}{\frac{{\cos}^{2} \left(t\right) + 1}{{\cos}^{2} \left(t\right)}} = \frac{2 \sin \left(t\right)}{{\cos}^{3} \left(t\right)}$

#### Explanation:

Try to simplify the expression.

$\frac{{\cos}^{2} \left(t\right) + 1}{{\cos}^{2} \left(t\right)} = 1 + {\sec}^{2} \left(t\right)$

Next, let $u = \sec \left(t\right)$.

$\frac{\text{d"u}{"d} t}{=} \sec \left(t\right) \tan \left(t\right)$

So,

$1 + {\sec}^{2} \left(t\right) = 1 + {u}^{2}$

Now, differentiate using the chain rule.

frac{"d"}{"d"t}(1+u^2) = frac{"d"}{"d"u}(1+u^2) * frac{"d"u}{"d"t}

$= 2 u \cdot \sec \left(t\right) \tan \left(t\right)$

$= 2 \sec \left(t\right) \cdot \sec \left(t\right) \tan \left(t\right)$

$= 2 {\sec}^{2} \left(t\right) \tan \left(t\right)$

$= \frac{2 \sin \left(t\right)}{{\cos}^{3} \left(t\right)}$