prove cos(x)=-1/2

Solution

prove

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

False

Solution steps

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

Show that the two sides are not equal

For

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

plug in

x = 0

cos (0) = 1

cos (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

The two sides are not equal

\Rightarrow False

\frac{1 + cos ( θ )}{tan ( θ ) + sin ( θ )} = cot (θ)

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

\frac{4 cot ( x )}{sec ( x )} = 4 csc (x) - 4 sin (x)

(tan (x) sec (- x)) cos (x) = sin (x) - 1

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