Is cos(3x)+cos(x)=2cos(2x)cos(x) ?

The answer to whether cos(3x)+cos(x)=2cos(2x)cos(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove cos(3x)+cos(x)=2cos(2x)cos(x)

prove

cos (3 x) + cos (x) = 2 cos (2 x) cos (x)

True

Solution steps

cos (3 x) + cos (x) = 2 cos (2 x) cos (x)

Manipulating left side

cos (3 x) + cos (x)

Rewrite using trig identities

cos (3 x) + cos (x)

Use the Sum to Product identity:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2})

Simplify

2 cos (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2}) : 2 cos (2 x) cos (x)

2 cos (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2})

\frac{3 x + x}{2} = 2 x

\frac{3 x + x}{2}

Add similar elements:

3 x + x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= 2 cos (2 x) cos (\frac{3 x - x}{2})

\frac{3 x - x}{2} = x

\frac{3 x - x}{2}

Add similar elements:

3 x - x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 cos (2 x) cos (x)

= 2 cos (2 x) cos (x)

= 2 cos (2 x) cos (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (4 x) = 2 sin (2 x) cos (2 x)

tan (x) csc^{2} (x) - tan (x) = cot (x)

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

sin (θ) + cos (θ) cot (θ) = csc (θ)

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1 = csc (x)

Is cos(3x)+cos(x)=2cos(2x)cos(x) ?
The answer to whether cos(3x)+cos(x)=2cos(2x)cos(x) is True