Is cos(2A)=1-2sin^2(A) ?

The answer to whether cos(2A)=1-2sin^2(A) is True

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prove cos(2A)=1-2sin^2(A)

prove

cos (2 A) = 1 - 2 sin^{2} (A)

True

Solution steps

cos (2 A) = 1 - 2 sin^{2} (A)

Manipulating left side

cos (2 A)

Rewrite using trig identities

cos (2 A)

Rewrite as

= cos (A + A)

Use the Angle Sum identity:

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

= cos (A) cos (A) - sin (A) sin (A)

Simplify

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

cos (A) cos (A) - sin (A) sin (A)

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

cos (A) cos (A)

Apply exponent rule:

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

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

= cos^{1 + 1} (A)

Add the numbers:

1 + 1 = 2

= cos^{2} (A)

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

sin (A) sin (A)

Apply exponent rule:

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

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

= sin^{1 + 1} (A)

Add the numbers:

1 + 1 = 2

= sin^{2} (A)

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

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

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

Use the Pythagorean identity:

cos^{2} (A) + sin^{2} (A) = 1

cos^{2} (A) = 1 - sin^{2} (A)

= 1 - sin^{2} (A) - sin^{2} (A)

Simplify

1 - sin^{2} (A) - sin^{2} (A) : 1 - 2 sin^{2} (A)

1 - sin^{2} (A) - sin^{2} (A)

Add similar elements:

- sin^{2} (A) - sin^{2} (A) = - 2 sin^{2} (A)

= 1 - 2 sin^{2} (A)

= 1 - 2 sin^{2} (A)

We showed that the two sides could take the same form

\Rightarrow True