Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
-$\cos x - \cos x \sin x = \cos ^ { 3 } x$
A) $\frac { \pi } { 4 }$
B) $\frac { \pi } { 2 }$
C) $\pi$
D) 0

Question

Quizplus · Accepted Answer

For $x = \frac{\pi}{4}$, the left side of the equation becomes $\cos\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{4}\right)$ which simplifies to $\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} - \frac{1}{2}$. The right side is $\cos^3\left(\frac{\pi}{4}\right) = \left(\frac{\sqrt{2}}{2}\right)^3 = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2} = \frac{1}{2\sqrt{2}}$. Since $\frac{\sqrt{2}}{2} - \frac{1}{2} \neq \frac{1}{2\sqrt{2}}$, the equation is not an identity for $x = \frac{\pi}{4}$.

Show That the Equation Is Not an Identity by Finding cos⁡x−cos⁡xsin⁡x=cos⁡3x\cos x - \cos x \sin x = \cos ^ { 3 } xcosx−cosxsinx=cos3x

Show That the Equation Is Not an Identity by Finding $\cos x - \cos x \sin x = \cos ^ { 3 } x$