# Question #01fe8

Dec 6, 2016

The equation cannot be verified. It is not true for all values of x.

#### Explanation:

An initial approach is to see if it is a trigonometric identity.
Starting with the left side and converting to sines and cosines,

$L S = S \in X C o s X C o t X$
$L S = S \in X C o s X \left(C o s \frac{X}{S} \in X\right)$
$L S = \left(\cancel{S} \in X\right) \cos X \left(C o s \frac{X}{\cancel{S} \in X}\right)$
$L S = C o {s}^{2} X$

The left side cannot equal the right side for all values of x since

$R S = C s c X$
$R S = \frac{1}{S} \in X$

To prove it is not an identity, it is sufficient to find one value for x which does not satisfy both sides. For example, working in degrees, if x = 30 degrees,

$L S = S \in \left(30\right) C o s \left(30\right) C o t \left(30\right)$
$L S = \left(0.5\right) \left(0.866\right) \left(1.732\right)$
$L S = 0.750 \left(\approx\right)$

$R S = C s c \left(30\right)$
$R S = 2$

Since the left side does not equal the right side for the value x = 30 degrees, the equation is not an identity and cannot be verified.