# How do you write (sec theta - 1)(sec theta + 1) in terms of sine and cosine?

Mar 6, 2018

Kindly refer to the Explanation.

#### Explanation:

Using the Identity : ${\sec}^{2} \theta = {\tan}^{2} \theta + 1$, we have,

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

Otherwise,

$\left(\sec \theta - 1\right) \left(\sec \theta + 1\right)$,

$= {\sec}^{2} \theta - 1$,

$= \frac{1}{\cos} ^ 2 \theta - 1$,

$= \frac{1 - {\cos}^{2} \theta}{\cos} ^ 2 \theta$,

$= {\sin}^{2} \frac{\theta}{\cos} ^ 2 \theta$, as above!

Mar 6, 2018

The expression in terms of sine and cosine is ${\sin}^{2} \frac{\theta}{\cos} ^ 2 \theta$.

#### Explanation:

First, you should multiply the expression and simplify as far as you can. Then, write everything in terms of sine and cosine.

Here are the identities we'll use:

color(white){color(black)( (sectheta=1/costheta, qquadqquad(1.1)), (sec^2theta=1/cos^2theta, qquadqquad(1.2)), (tantheta=sintheta/costheta, qquadqquad(2.1)), (tan^2theta=sin^2theta/cos^2theta, qquadqquad(2.2)), (sin^2theta+cos^2theta=1, qquadqquad(3.1)), (sin^2theta/cos^2theta+cos^2theta/cos^2theta=1/cos^2theta, qquadqquad(3.2)), (tan^2theta+1=sec^2theta, qquadqquad(3.3)):}

Some notes: identity $\left(1.2\right)$ was achieved by squaring both sides of identity $\left(1.1\right)$ (same with $\left(2.2\right)$ and $\left(2.1\right)$).

Similarly, identity $\left(3.2\right)$ was achieved by dividing all the terms in identity $\left(3.1\right)$ by ${\cos}^{2} \theta$. Then, identity $\left(3.3\right)$ was reached by simplifying identity $\left(3.2\right)$ using previously-proved identities $\left(1.2\right)$ and $\left(2.2\right)$

Now, here's the expression:

color(white){color(black)( ((sectheta-1)(sectheta+1), qquadqquad"The problem"), (sec^2theta+sectheta-sectheta-1, qquadqquad"Multiplying out the expression"), (sec^2thetacolor(red)cancelcolor(black)(+sectheta-sectheta)-1, qquadqquad"Like terms cancel out"), (sec^2theta-1, qquadqquad"Rewrite the above step"), (tan^2theta+1-1, qquadqquad"Replace "sec^2theta" with "tan^2theta+1" using identity "(3.3)), (tan^2thetacolor(red)cancelcolor(black)(+1-1), qquadqquad"Like terms cancel out"), (tan^2theta, qquadqquad "Rewrite the above step"), (sin^2theta/cos^2theta, qquadqquad "Replace "tan^2theta" with "sin^2theta/cos^2theta" using identity "(2.2)):}

That's the answer. Hope this helped!