Is 1+cos(2x)=2cos^2(x) ?

The answer to whether 1+cos(2x)=2cos^2(x) is True

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prove 1+cos(2x)=2cos^2(x)

prove

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

True

Solution steps

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

Manipulating left side

1 + cos (2 x)

Rewrite using trig identities

1 + cos (2 x)

Use the Double Angle identity:

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

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

Simplify

1 + 2 cos^{2} (x) - 1 : 2 cos^{2} (x)

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

= 2 cos^{2} (x)

= 2 cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (x) (csc (x) - sin (x)) = cos (x)

tan (x) (sin (x) + cot (x) cos (x)) = sec (x)

sin (x) cos^{2} (x) + sin^{3} (x) = sin (x)

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

cos (\frac{π}{3}) = \frac{1}{2}