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

Short Answer

Expert verified

Hence, proved.

Step by step solution

Part (1) Step 1. Given Information.

The given pythagoaras identity is :

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

Part (1) Step 2. Proving the identity.

Use Pythagorean identity,

$\sin^{2} x + \cos^{2} x = 1 \cos^{2} x = 1 - \sin^{2} x$

Then,

$\cos 2 x = \cos^{2} x - \sin^{2} x$

To substitute into the first double-angle,

$\begin{array}{rcl} \cos 2 x & = & (1 - \sin^{2} x) - \sin^{2} x \\ \cos 2 x & = & 1 - 2 \sin^{2} x \\ \sin^{2} x & = & \frac{1}{2} (1 - \cos 2 x) \end{array}$

Part (2) Step 1. Proving the identity.

Take the Pythagorean equation in this form, $\sin^{2} x = 1 - \cos^{2} x$ and substitute with the First double-angle identity.

\sin^{2} x + \cos^{2} x = 1 1 - \cos^{2} x = \sin^{2} x

$\cos 2 x = \cos^{2} x - \sin^{2} x$

To substitute into the first double-angle,

$\begin{array}{rcl} \cos 2 x & = & - (1 - \cos^{2} x) + co s^{2} x \\ \cos 2 x & = & - 1 + 2 \cos^{2} x \\ \cos^{2} x & = & \frac{1}{2} (1 + \cos 2 x) \end{array}$ .

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

Short Answer

Step by step solution

Part (1) Step 1. Given Information.

Part (1) Step 2. Proving the identity.

Part (2) Step 1. Proving the identity.

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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−cos2x)2.cos2x=12(1+cos2x).

Short Answer

Step by step solution

Part (1) Step 1. Given Information.

Part (1) Step 2. Proving the identity.

Part (2) Step 1. Proving the identity.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Calculus

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Decision Maths

Study anywhere. Anytime. Across all devices.

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