Is cos^2(x)= 1/2 (1+cos(2x)) ?

The answer to whether cos^2(x)= 1/2 (1+cos(2x)) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove cos^2(x)= 1/2 (1+cos(2x))

prove

cos^{2} (x) = \frac{1}{2} (1 + cos (2 x))

True

Solution steps

cos^{2} (x) = \frac{1}{2} (1 + cos (2 x))

Manipulating right side

\frac{1}{2} (1 + cos (2 x))

Rewrite using trig identities

\frac{1}{2} (1 + cos (2 x))

Use the Double Angle identity:

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

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

Simplify

\frac{1}{2} (1 + 2 cos^{2} (x) - 1) : cos^{2} (x)

\frac{1}{2} (1 + 2 cos^{2} (x) - 1)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot ( 1 + 2 cos ^{2} ( x ) - 1 )}{2}

1 \cdot (1 + 2 cos^{2} (x) - 1) = 2 cos^{2} (x)

1 \cdot (1 + 2 cos^{2} (x) - 1)

Simplify

1 + 2 cos^{2} (x) - 1 : 2 cos^{2} (x)

1 + 2 cos^{2} (x) - 1

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

= 1 \cdot 2 cos^{2} (x)

Multiply the numbers:

1 \cdot 2 = 2

= 2 cos^{2} (x)

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

Divide the numbers:

\frac{2}{2} = 1

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (2 x) = sin (x + x)

8 cos^{4} (x) - 8 cos^{2} (x) + 1 = cos (4 x)

\frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) sin ( 4 x )} = tan (x)

\frac{tan ( x ) + cot ( x )}{csc ( x )} = sec (x)

\frac{csc ( x )}{cot ( x )} = sec (x)

Is cos^2(x)= 1/2 (1+cos(2x)) ?
The answer to whether cos^2(x)= 1/2 (1+cos(2x)) is True