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

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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

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