CK-12-Pre-Calculus Concepts

6.4. Double, Half, and Power Reducing Identities http://www.ck12.org

• cos^2 x=^1 +cos 2 2 x


• tan^2 x=^11 −+cos 2cos 2xx

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

• sin 2 x=±

√

1 −cosx

2

• cosx 2 =±

√

1 +cosx

2

• tanx 2 =±

√ 1 −cosx

1 +cosx

Example A
Rewrite sin^4 xas an expression without powers greater than one.
Solution: While sinx·sinx·sinx·sinxdoes technically solve this question, try to get the terms to sum together not
multiply together.

sin^4 x= (sin^2 x)^2

=

( 1 −cos 2x

2

) 2

=^1 −2 cos 2x+cos

(^22) x
4
=^14

(

1 −2 cos 2x+^1 +cos 4 2 x

)

Example B
Write the following expression with only sinxand cosx: sin 2x+cos 3x.
Solution:

sin 2x+cos 3x=2 sinxcosx+cos( 2 x+x)

=2 sinxcosx+cos 2xcosx−sin 2xsinx

=2 sinxcosx+(cos^2 x−sin^2 x)cosx−(2 sinxcosx)sinx

=2 sinxcosx+cos^3 x−sin^2 xcosx−2 sin^2 xcosx

=2 sinxcosx+cos^3 x−3 sin^2 xcosx

Example C
Use half angles to find an exact value of tan 22. 5 ◦without using a calculator.

Solution:tanx 2 =±

√ 1 −cosx

1 +cosx

tan 22. 5 ◦=tan^452 ◦=±

√

1 −cos 45◦

1 +cos 45◦=±

√√

√√ 1 −√ 22

1 +

√ 2

2

=±

√√

√√^22 −√ 22

(^22) +√ 22 =±

√

2 −

√

2

2 +√ 2

Sometimes you may be requested to get all the radicals out of the denominator.