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Chapter 5: Q. 2 TB. (page 451)

Double-angle identities: Prove each of the following double - angle identities by applying the sum identity for the cosine followed by a Pythagorean identity.

1. sin2x=12(1鈭�cos2x)

2.cos2x=12(1+cos2x).

Short Answer

Expert verified

Hence, proved.

Step by step solution

01

Part (1)  Step 1. Given Information.

The given pythagoaras identity is :

sin2x+cos2x=1.

02

Part (1)  Step 2. Proving the identity.

Use Pythagorean identity,

sin2x+cos2x=1cos2x=1鈥�sin2x

Then,

cos2x=cos2x鈥�sin2x

To substitute into the first double-angle,

cos2x=(1鈥�sin2x)鈥�sin2xcos2x=1鈥�2sin2xsin2x=121-cos2x

03

Part (2) Step 1.  Proving the identity.

Take the Pythagorean equation in this form, sin2x=1鈥�cos2xand substitute with the First double-angle identity.

sin2x+cos2x=11-cos2x=sin2x

cos2x=cos2x鈥�sin2x

To substitute into the first double-angle,

cos2x=-(1鈥�cos2x)+cos2xcos2x=-1+2cos2xcos2x=121+cos2x.

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