Is cos(2α)=cos(α+α) ?

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

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prove cos(2α)=cos(α+α)

prove

cos (2 α) = cos (α + α)

True

Solution steps

cos (2 α) = cos (α + α)

Manipulating left side

cos (2 α)

Rewrite using trig identities

cos (2 α)

Use the Double Angle identity:

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

= cos^{2} (α) - sin^{2} (α)

= cos^{2} (α) - sin^{2} (α)

Manipulating right side

cos (α + α)

Rewrite using trig identities

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)

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)

= cos^{2} (α) - sin^{2} (α)

We showed that the two sides could take the same form

\Rightarrow True

cos^{4} (x) - sin^{4} (x) = 2 tan^{2} (x) - 1

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

- 3 (0) cos (3 t) + 3 (1) sin (3 t) = 3 sin (3 t)

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

cot^{2} (x) = csc^{2} (- 1)