Is cos(2x)=1-2sin^2(x) ?

The answer to whether cos(2x)=1-2sin^2(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove cos(2x)=1-2sin^2(x)

prove

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

True

Solution steps

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

Manipulating left side

cos (2 x)

Rewrite using trig identities

cos (2 x)

Rewrite as

= cos (x + x)

Use the Angle Sum identity:

cos (s + s) = cos (s) cos (s) - sin (s) sin (s)

= cos (x) cos (x) - sin (x) sin (x)

Simplify

cos (x) cos (x) - sin (x) sin (x) : cos^{2} (x) - sin^{2} (x)

cos (x) cos (x) - sin (x) sin (x)

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

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

= 1 - sin^{2} (x) - sin^{2} (x)

Simplify

1 - sin^{2} (x) - sin^{2} (x) : 1 - 2 sin^{2} (x)

1 - sin^{2} (x) - sin^{2} (x)

Add similar elements:

- sin^{2} (x) - sin^{2} (x) = - 2 sin^{2} (x)

= 1 - 2 sin^{2} (x)

= 1 - 2 sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True