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Simplifying Trig Expressions using Double and Half Angle Formulas

As Agent Trigonometry, you are given the following cryptic clue. How could you simplify this clue?

$\frac{\tan 2x}{\frac{tan x}{1 + \tan x}}$

Guidance

We can also use the double-angle and half-angle formulas to simplify trigonometric expressions.

Example A

Simplify $\frac{\cos 2x}{\sin x \cos x}$ .

Solution: Use $\cos 2a=\cos^2a-\sin^2a$ and then factor.

$\frac{\cos 2x}{\sin x \cos x}&=\frac{\cos^2x- \sin^2x}{\sin x+ \cos x} \\&=\frac{\left(\cos x- \sin x\right) \cancel{\left(\cos x + \sin x\right)}}{\cancel{\sin x+ \cos x}} \\&=\cos x- \sin x$

Example B

Find the formula for $\sin 3x$ .

Solution: You will need to use the sum formula and the double-angle formula. $\sin 3x=\sin(2x+x)$

$\sin 3x&=\sin (2x+x) \\&=\sin 2x \cos x + \cos 2x \sin x \\&=2 \sin x \cos x \cos x+ \sin x(2 \cos^2x-1) \\&=2 \sin x \cos^2x+2 \sin x \cos^2 x- \sin x \\&=4 \sin x \cos^2x- \sin x \\&=\sin x(4 \cos^2x-1)$

We will explore other possibilities for the $\sin 3x$ because of the different formulas for $\cos 2a$ in the Problem Set.

Example C

Verify the identity $\cos x+2 \sin^2 \frac{x}{2}=1$ .

Solution: Simplify the left-hand side use the half-angle formula.

$& \cos x+2 \sin^2 \frac{x}{2} \\& \cos x+2 \left(\sqrt{\frac{1- \cos x}{2}}\right)^2 \\& \cos x+2 \cdot \frac{1- \cos x}{2} \\& \cos x+1- \cos x \\& 1$

Concept Problem Revisit

Use $\tan 2a=\frac {2\tan a}{1 - \tan^2 a}$ and then factor.

$\frac{\tan 2x}{\frac{tan x}{1 + \tan x}}=\frac{2\tan x}{1-\tan^2 x}\cdot \frac{1 + \tan x}{tan x} \\=\frac{2\tan x}{(1 + \tan x)(1 - \tan x)}\cdot \frac{1 + \tan x}{tan x} =\frac{2}{1-\tan x}$

Guided Practice

1. Simplify $\frac{\sin 2x}{\sin x}$ .

2. Verify $\cos x+2 \cos^2 \frac{x}{2}=1+ 2 \cos x$ .

1. $\frac{\sin 2x}{\sin x}=\frac{2 \sin x \cos x}{\sin x}=2 \cos x$

2. $\cos x+2 \cos^2 \frac{x}{2}&=1+2 \cos x \\\cos x+2 \sqrt{\frac{1+ \cos x}{2}}^2&= \\\cos x+1 + \cos x&= \\1+2 \cos x&=$

Practice

Simplify the following expressions.

1. $\sqrt{2+2 \cos x} \left(\cos \frac{x}{2}\right)$
2. $\frac{\cos 2x}{\cos^2x}$
3. $\tan 2x(1+ \tan x)$
4. $\cos 2x- 3 \sin^2x$
5. $\frac{1+\cos 2x}{\cot x}$
6. $(1+ \cos x)^2 \tan \frac{x}{2}$

Verify the following identities.

1. $\cot \frac{x}{2}=\frac{\sin x}{1- \cos x}$
2. $\frac{\sin x}{1+ \cos x}=\frac{1- \cos x}{\sin x}$
3. $\frac{\sin 2x}{1+ \cos 2x}= \tan x$
4. $(\sin x+ \cos x)^2=1+ \sin 2x$
5. $\sin x \tan \frac{x}{2}+2 \cos x=2 \cos^2 \frac{x}{2}$
6. $\cot x+ \tan x=2 \csc 2x$
7. $\cos 3x=4 \cos^3x-3 \cos x$
8. $\cos 3x=\cos^3x-3 \sin^2x \cos x$
9. $\sin 2x-\tan x=\tan x \cos 2x$
10. $\cos^4x-\sin^4x=\cos 2x$