6.4: Double, Half, and Power Reducing Identities

These identities are significantly more involved and less intuitive than previous identities. By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase. Each identity in this concept is named aptly. Double angles work on finding \(\sin 80^{\circ}\) if you already know \(\sin 40^{\circ}\). Half angles allow you to find \(\sin 15^{\circ}\) if you already know \(\sin 30^{\circ}\). Power reducing identities allow you to find \(\sin ^{2} 15^{\circ}\) if you know the sine and cosine of \(30^{\circ}\)

What is \(\sin ^{2} 15^{\circ} ?\)

Double Angle, Half Angle, and Power Reducing Identities

Double Angle Identities

The double angle identities are proved by applying the sum and difference identities. They are left as review problems. These are the double angle identities.

• \(\sin 2 x=2 \sin x \cos x\)
• \(\cos 2 x=\cos ^{2} x-\sin ^{2} x\)
• \(\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}\)

Half Angle Identities

The half angle identities are a rewritten version of the power reducing identities. The proofs are left as review problems.

• \(\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}\)
• \(\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}\)
• \(\tan \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}\)

Power Reducing Identities

The power reducing identities allow you to write a trigonometric function that is squared in terms of smaller powers. The proofs are left as examples and review problems.

• \(\sin ^{2} x=\frac{1-\cos 2 x}{2}\)
• \(\cos ^{2} x=\frac{1+\cos 2 x}{2}\)
• \(\tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x}\)

Power reducing identities are most useful when you are asked to rewrite expressions such as \(\sin ^{4} x\) as an expression without powers greater than one. While \(\sin x \cdot \sin x \cdot \sin x \cdot \sin x\) does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.

\(\begin{aligned} \sin ^{4} x &=\left(\sin ^{2} x\right)^{2} \\ &=\left(\frac{1-\cos 2 x}{2}\right)^{2} \\ &=\frac{1-2 \cos 2 x+\cos ^{2} 2 x}{4} \\ &=\frac{1}{4}\left(1-2 \cos 2 x+\frac{1+\cos 4 x}{2}\right) \end{aligned}\)

Examples

Example 1

Earlier, you were asked to find \(\sin ^{2} 15^{\circ}\). In order to fully identify \(\sin ^{2} 15^{\circ}\) you need to use the power reducing formula.

\(
\begin{aligned}
\sin ^{2} x &=\frac{1-\cos 2 x}{2} \\
\sin ^{2} 15^{\circ} &=\frac{1-\cos 30^{\circ}}{2}=\frac{1}{2}-\frac{\sqrt{3}}{4} \\
&=\frac{2-\sqrt{3}}{4}
\end{aligned}
\)

Example 2

Write the following expression with only \(\sin x\) and \(\cos x: \sin 2 x+\cos 3 x\).

\(
\begin{aligned}
\sin 2 x+\cos 3 x &=2 \sin x \cos x+\cos (2 x+x) \\
&=2 \sin x \cos x+\cos 2 x \cos x-\sin 2 x \sin x \\
&=2 \sin x \cos x+\left(\cos ^{2} x-\sin ^{2} x\right) \cos x-(2 \sin x \cos x) \sin x \\
&=2 \sin x \cos x+\cos ^{3} x-\sin ^{2} x \cos x-2 \sin ^{2} x \cos x \\
&=2 \sin x \cos x+\cos ^{3} x-3 \sin ^{2} x \cos x
\end{aligned}
\)

Example 3

Use half angles to find an exact value of tan \(22.5^{\circ}\) without using a calculator.

\(
\begin{array}{l}
\tan \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}} \\
\qquad \begin{aligned}
\tan 22.5^{\circ} &=\tan \frac{45^{\circ}}{2}=\pm \sqrt{\frac{1-\cos 45^{\circ}}{1+\cos 45^{\circ}}}=\pm \sqrt{\frac{1-\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}}=\pm \sqrt{\frac{\frac{2}{2}-\frac{\sqrt{2}}{2}}{\frac{2}{2}+\frac{\sqrt{2}}{2}}}=\pm \sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}} \\
&=\pm \sqrt{\frac{(2-\sqrt{2})^{2}}{2}}
\end{aligned}
\end{array}
\)

Example 4

Prove the power reducing identity for sine.

\(\sin ^{2} x=\frac{1-\cos 2 x}{2}\)

Using the double angle identity for cosine:
\(
\begin{array}{l}
\cos 2 x=\cos ^{2} x-\sin ^{2} x \\
\cos 2 x=\left(1-\sin ^{2} x\right)-\sin ^{2} x \\
\cos 2 x=1-2 \sin ^{2} x
\end{array}
\)

This expression is an equivalent expression to the double angle identity and is often considered an alternate form.

Example 5

Simplify the following identity. \(\sin ^{4} x-\cos ^{4} x\).

Here are the steps:
\(
\begin{aligned}
\sin ^{4} x-\cos ^{4} x &=\left(\sin ^{2} x-\cos ^{2} x\right)\left(\sin ^{2} x+\cos ^{2} x\right) \\
&=-\left(\cos ^{2} x-\sin ^{2} x\right) \\
&=-\cos 2 x
\end{aligned}
\)

Review

Prove the following identities.

1. \(\sin 2 x=2 \sin x \cos x\)

2. \(\cos 2 x=\cos ^{2} x-\sin ^{2} x\)

3. \(\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}\)

4. \(\cos ^{2} x=\frac{1+\cos 2 x}{2}\)

5. \(\tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x}\)

6. \(\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}\)

7. \(\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}\)

8. \(\tan \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}\)

9. \(\csc 2 x=\frac{1}{2} \csc x \sec x\)

10. \(\cot 2 x=\frac{\cot ^{2} x-1}{2 \cot x}\)

Find the value of each expression using half angle identities.

11. \(\tan 15^{\circ}\)

12. \(\tan 22.5^{\circ}\)

13. \(\sec 22.5^{\circ}\)

14. Show that \(\tan \frac{x}{2}=\frac{1-\cos x}{\sin x}\)

15. Using your knowledge from the answer to question \(14,\) show that \(\tan \frac{x}{2}=\frac{\sin x}{1+\cos x}\).