prove sin(x)= 1/2

Solution

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

x = 0

sin (0) = 0

sin (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

= 0

The two sides are not equal

\Rightarrow False

\frac{2 sin ( x ) cos ( x )}{2 sin ^{2} ( x )} = cot (x)

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

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

- cot (\frac{x}{2}) = - (csc (x) + cot (x))

1 + tan^{2} (y) = sec^{2} (y)