Is cos(2u)=cos^2(u)-sin^2(u) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove cos(2u)=cos^2(u)-sin^2(u)

prove

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

True

Solution steps

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

Manipulating left side

cos (2 u)

Rewrite using trig identities

cos (2 u)

Rewrite as

= cos (u + u)

Use the Angle Sum identity:

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

= cos (u) cos (u) - sin (u) sin (u)

= cos (u) cos (u) - sin (u) sin (u)

Simplify

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

cos (u) cos (u) - sin (u) sin (u)

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

cos (u) cos (u)

Apply exponent rule:

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

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

= cos^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= cos^{2} (u)

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

sin (u) sin (u)

Apply exponent rule:

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

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

= sin^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= sin^{2} (u)

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e (csc (θ) + 1) (csc (θ) - 1) = cot^{2} (θ)

p ro v e cos (θ) (cot (θ) + tan (θ)) = csc (θ)

p ro v e sin (- θ) = - sin (θ)

p ro v e - sin (x) = sin (- x)

p ro v e \frac{sin ( y ) + tan ( y )}{1 + sec ( y )} = sin (y)

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