### Home > APCALC > Chapter 4 > Lesson 4.5.1 > Problem4-175

4-175.

Rewrite each of the following expressions using a single trigonometric function. You may wish to review your trigonometric identities from Chapter 1.

Common Trigonometric Identities

 Reciprocal  $\sec(θ)=\frac{1}{\cos(\theta)}$  $\csc(θ)=\frac{1}{\sin(\theta)}$ $\tan(θ)=\frac{\sin(\theta)}{\cos(\theta)}$  $\cot(θ)=\frac{\cos(\theta)}{\sin(\theta)}$ Pythagorean $\sin^2\left(θ\right)+\cos^2\left(θ\right)=1$ $\tan^2\left(θ\right)+1=\sec^2\left(θ\right)$ $1+\cot^2\left(θ\right)=\csc^2\left(θ\right)$ Angle Sum $\sin\left(a\pm b\right)=\sin\left(a\right)\cos\left(b\right)\pm\cos\left(a\right)\sin\left(b\right)$ $\cos\left(a\pm b\right)=\cos\left(a\right)\cos\left(b\right)∓\sin\left(a\right)\sin\left(b\right)$ Double Angle $\sin\left(2a\right)=2\sin\left(a\right)\cos\left(a\right)$ $\operatorname { cos } ( 2 a ) = \left\{ \begin{array} { l } { \operatorname { cos } ^ { 2 } ( a ) - \operatorname { sin } ^ { 2 } ( a ) } \\ { 2 \operatorname { cos } ^ { 2 } ( a ) - 1 } \\ { 1 - 2 \operatorname { sin } ^ { 2 } ( a ) } \end{array} \right.$

1. $10\sin\left(3x\right)\cos\left(3x\right)$

Double Angle identity (factor first).

2. $\sin(x)\cos(3x)-\sin(3x)\cos(x)$

Sum and Difference (Angle Sum) identity.

3. $\cos^4\left(x\right)-\sin^4\left(x\right)$

Factor first.

4. $\tan(x)+\cot(x)$

Start by rewriting $\tan(x)$ and $\cot(x)$ as fractions.

You will use more than one identity.

$\frac{\sin(x)}{\cos(x)}+\frac{\cos(x)}{\sin(x)}=\frac{\sin^2(x)+\cos^2(x)}{\cos(x)\sin(x)}=$

$\frac{1}{\frac{1}{2}\text{sin}(2x)}=2\text{csc}(2x)$