Is cos(2α)=cos^2(α)-sin^2(α) ?

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

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prove cos(2α)=cos^2(α)-sin^2(α)

prove

cos (2 α) = cos^{2} (α) - sin^{2} (α)

True

Solution steps

cos (2 α) = cos^{2} (α) - sin^{2} (α)

Manipulating left side

cos (2 α)

Rewrite using trig identities

cos (2 α)

Rewrite as

= cos (α + α)

Use the Angle Sum identity:

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

= cos (α) cos (α) - sin (α) sin (α)

= cos (α) cos (α) - sin (α) sin (α)

Simplify

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

cos (α) cos (α) - sin (α) sin (α)

cos (α) cos (α) = cos^{2} (α)

cos (α) cos (α)

Apply exponent rule:

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

cos (α) cos (α) = cos^{1 + 1} (α)

= cos^{1 + 1} (α)

Add the numbers:

1 + 1 = 2

= cos^{2} (α)

sin (α) sin (α) = sin^{2} (α)

sin (α) sin (α)

Apply exponent rule:

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

sin (α) sin (α) = sin^{1 + 1} (α)

= sin^{1 + 1} (α)

Add the numbers:

1 + 1 = 2

= sin^{2} (α)

= cos^{2} (α) - sin^{2} (α)

= cos^{2} (α) - sin^{2} (α)

We showed that the two sides could take the same form

\Rightarrow True

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

2 csc (2 x) = \frac{1}{sin ( x ) cos ( x )}

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

cos^{2} (θ) - sin^{2} (θ) = 1

\frac{cos ( θ )}{1 - sin ^{2} ( θ )} = \frac{1}{cos ( θ )}

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