# How do you prove sinx/cscx + cosx/secx = sinx cscx?

##### 2 Answers

TRUE,$\text{ " " } \sin \frac{x}{\csc} x + \cos \frac{x}{\sec} x = \sin x \csc x$

#### Explanation:

Take note first that $\sin x \csc x = 1$

Why? because, $\sin x = \frac{1}{\csc} x$

multiplying both sides by csc x

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

$\sin x \cdot \csc x = \frac{1}{\cancel{\csc}} x \cdot \cancel{\csc} x$

and

$\sin x \csc x = 1$

Take note also that

$\sec x \cos x = 1$ because $\sec x = \frac{1}{\cos} x$

Let us go back to the problem

$\sin \frac{x}{\csc} x + \cos \frac{x}{\sec} x = \sin x \csc x$

$\sin \frac{x}{\csc} x \cdot \sin \frac{x}{\sin} x + \cos \frac{x}{\sec} x \cdot \cos \frac{x}{\cos} x = \sin x \csc x$

because $\sin \frac{x}{\sin} x = 1$ and $\cos \frac{x}{\cos} x = 1$

You can always multiply any quantity by 1 and will not change anything

$\sin \frac{x}{\csc} x \cdot \sin \frac{x}{\sin} x + \cos \frac{x}{\sec} x \cdot \cos \frac{x}{\cos} x = \sin x \csc x$

Next

${\sin}^{2} \frac{x}{\csc x \sin x} + {\cos}^{2} \frac{x}{\sec x \cos x} = \sin x \csc x$

${\sin}^{2} \frac{x}{1} + {\cos}^{2} \frac{x}{1} = \sin x \csc x$

${\sin}^{2} x + {\cos}^{2} x = \sin x \csc x$

Also from Pythagorean Relation,

${\sin}^{2} x + {\cos}^{2} x = 1$

therefore

${\sin}^{2} x + {\cos}^{2} x = \sin x \csc x$

becomes

$1 = \sin x \csc x$

$\sin x \csc x = \sin x \csc x$ TRUE !!!

God bless America ...

Another way is

$\sin \frac{x}{\csc} x + \cos \frac{x}{\sec} x = \sin x \csc x$

Starting from left, factor out $\sin x$

$\sin x \left(\frac{1}{\csc} x + \cos \frac{x}{\sin x \sec x}\right) = \sin x \csc x$

$\sin x \left(\frac{1}{\csc} x + \cos \frac{x}{\sin} x \cdot \frac{1}{\sec} x\right) = \sin x \csc x$

From reciprocal relations: $\frac{1}{\csc} x = \sin x$ and $\frac{1}{\sec} x = \cos x$
it follows

$\sin x \left(\sin x + \cos \frac{x}{\sin} x \cdot \cos x\right) = \sin x \csc x$

and then

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

Simplify by using $\sin x$ as Least Common Denominator

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

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

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

$\sin x \csc x = \sin x \csc x$

Proven !!