prove cos^3(θ)=-1

Solution

prove

cos^{3} (θ) = - 1

False

Solution steps

cos^{3} (θ) = - 1

Show that the two sides are not equal

For

cos^{3} (θ) = - 1

plug in

θ = 0

cos^{3} (0) = 1

cos^{3} (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^{3}

Simplify

= 1

The two sides are not equal

\Rightarrow False

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

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

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

cos (k (x) x) = cos (- k (x) x)

tan^{2} (x) = sin^{2} (x) + sin^{2} (x) tan^{2} (x)