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

The answer to whether 1-2cos^2(x)=2sin^2(x) is False

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

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

x = 0

1 - 2 cos^{2} (0) = - 1

1 - 2 cos^{2} (0)

Use the following trivial identity:

cos (0) = 1

cos (0)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

= 1 - 2 \cdot 1^{2}

Simplify

= - 1

2 sin^{2} (0) = 0

2 sin^{2} (0)

Use the following trivial identity:

sin (0) = 0

sin (0)

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 2 \cdot 0^{2}

Simplify

= 0

The two sides are not equal

\Rightarrow False

sin (\frac{2 π}{3}) = \frac{3}{2}

\frac{1}{2} 2 sin (x) cos (x) = \frac{1}{2} sin (2 x)

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

\frac{cot ( x ) + 1}{csc ( x )} = cos (x) + sin (x)

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