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Simplifying Trigonometric Expressions with Double-Angle Identities

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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 .

Answers

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

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