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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

True

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

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

sin (y) tan (y) = cos (y)

sec^{4} (\frac{x}{3}) = \frac{( sec ^{4} ( x ) )}{3}

2 sin (z) cos (z) = sin (2 z)

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

- tan (x) cos (x) = tan (x)

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