The answer to whether sin(2θ)= 1/2 is False

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prove sin(2θ)= 1/2

Solution

prove

sin (2 θ) = \frac{1}{2}

False

Solution steps

sin (2 θ) = \frac{1}{2}

Show that the two sides are not equal

For

sin (2 θ) = \frac{1}{2}

plug in

θ = 0

sin (2 \cdot 0) = 0

sin (2 \cdot 0)

Simplify

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

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

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

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

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