Is cos x sin x = 1 an identity?

Feb 23, 2016

No

Explanation:

An easy way to show that this is not an identity is to plug in $0$ for $x$.

$\cos \left(0\right) \sin \left(0\right) = 0 \cdot 1 = 0 \ne 1$

In fact, if we use the identity $\sin \left(2 x\right) = 2 \sin \left(x\right) \cos \left(x\right)$ we can show that there are no real values for which the questioned equality is true.

$\cos \left(x\right) \sin \left(x\right) = \frac{1}{2} \left(2 \sin \left(x\right) \cos \left(x\right)\right) = \sin \frac{2 x}{2}$

Because $\sin \left(2 x\right) \le 1$ for all $x \in \mathbb{R}$ (for all real-valued $x$) that means that $\cos \left(x\right) \sin \left(x\right) \le \frac{1}{2}$ for all real $x$.

The more common trig identity which involves $1$ is

${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$

which is true for all $x$.

Feb 23, 2016

Absolutely not

Explanation:

According to this

$\cos x = \frac{1}{\sin} x = \csc x$

\ I hope you get the point IT IS NOT TRUE FOR ALL VALUES OF X

So it is not an identity