1. ## translating secant

can someone explain this step? I understand why the secx in the numerator divides with sec^2x to give just secx in the denominator but what does sec^2x or tan^2x translate into that will give the numerator expression of 1+tanx?

(perhaps this should be posted in pre-calc? but it's the end bit of a calculus problem involving taking the derivative of an expression with trig functions)

2. ## Re: translating secant

From the Pythagorean Identity

\displaystyle \displaystyle \begin{align*} \sin^2{x} + \cos^2{x} &\equiv 1 \\ \frac{\sin^2{x} + \cos^2{x}}{\cos^2{x}} &\equiv \frac{1}{\cos^2{x}} \\ \frac{\sin^2{x}}{\cos^2{x}} + \frac{\cos^2{x}}{\cos^2{x}} &\equiv \sec^2{x} \\ \tan^2{x} + 1 &\equiv \sec^2{x} \\ 1 &\equiv \sec^2{x} - \tan^2{x} \end{align*}

So that means in your example

\displaystyle \displaystyle \begin{align*} \frac{\sec{x}\left( \sec^2{x} - \tan^2{x} + \tan{x} \right)}{\sec^2{x}} &\equiv \frac{\sec{x} \left( 1 + \tan{x} \right)}{\sec^2{x}} \\ &\equiv \frac{1 + \tan{x}}{\sec{x}} \end{align*}

thanks!