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

sec^{2} (x) - tan^{2} (x) = tan (x) cot (x)

\frac{tan ( θ )}{tan ( θ ) - 1} = \frac{1}{1 - cot ( θ )}

csc (u) - sin (u) = cos (u) cot (u)

\frac{tan ^{2} ( x )}{sec ( x ) - 1} = sec (x) + 1

\frac{1 + tan ^{2} ( x )}{csc ( x ) sec ( x )} = tan (x)