Is cos(2t)=cos^2(t)-sin^2(t) ?

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

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prove cos(2t)=cos^2(t)-sin^2(t)

prove

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

True

Solution steps

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

Manipulating left side

cos (2 t)

Rewrite using trig identities

cos (2 t)

Rewrite as

= cos (t + t)

Use the Angle Sum identity:

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

= cos (t) cos (t) - sin (t) sin (t)

= cos (t) cos (t) - sin (t) sin (t)

Simplify

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

cos (t) cos (t) - sin (t) sin (t)

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

cos (t) cos (t)

Apply exponent rule:

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

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

= cos^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= cos^{2} (t)

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

sin (t) sin (t)

Apply exponent rule:

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

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

= sin^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= sin^{2} (t)

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

sec^{2} (B) - csc^{2} (B) = \frac{tan ( B ) - cot ( B )}{sin ( B ) cos ( B )}

2 + sec (a) cos (a) = 3

5 cos^{2} (θ) + 2 sin^{2} (θ) = 3 cos^{2} (θ) + 2

\frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

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