Is sin^2(θ)=(1-cos(2θ))/2 ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove sin^2(θ)=(1-cos(2θ))/2

prove

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

True

Solution steps

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

Manipulating right side

\frac{1 - cos ( 2 θ )}{2}

Rewrite using trig identities

\frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity:

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

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

Simplify

\frac{1 - ( 1 - 2 sin ^{2} ( θ ) )}{2} : sin^{2} (θ)

\frac{1 - ( 1 - 2 sin ^{2} ( θ ) )}{2}

Expand

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

1 - (1 - 2 sin^{2} (θ))

- (1 - 2 sin^{2} (θ)) : - 1 + 2 sin^{2} (θ)

- (1 - 2 sin^{2} (θ))

Distribute parentheses

= - (1) - (- 2 sin^{2} (θ))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + 2 sin^{2} (θ)

= 1 - 1 + 2 sin^{2} (θ)

1 - 1 = 0

= 2 sin^{2} (θ)

= \frac{2 sin ^{2} ( θ )}{2}

Divide the numbers:

\frac{2}{2} = 1

= sin^{2} (θ)

= sin^{2} (θ)

= sin^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

sin (3 x) = 4 sin (x) cos^{2} (x) - sin (x)

(1 + cot (x))^{2} = csc^{2} (x) + 2 cot (x)

sec (x) - tan (x) sin (x) = cos (x)

sin (θ) cot (θ) = cos (θ)

tan (θ) csc (θ) cos (θ) = 1

Is sin^2(θ)=(1-cos(2θ))/2 ?
The answer to whether sin^2(θ)=(1-cos(2θ))/2 is True