How do you verify the identify sintheta/csctheta=sin^2theta?

Mar 7, 2018

$L H S \implies \sin \frac{x}{\csc} x$

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

$\implies \sin x \cdot \sin x = {\sin}^{2} x$ since, $\sin x = \frac{1}{\csc} x$

Mar 7, 2018

$\text{please look at the following explanation.}$

Explanation:

$\csc = \frac{1}{\sin} \theta$

$\text{write "1/sin theta " instead " csc theta " in the equation } \sin \frac{\theta}{\csc} \theta$

$\sin \frac{\theta}{\frac{1}{\sin} \theta} = \sin \theta \cdot \sin \frac{\theta}{1} = {\sin}^{2} \theta$

See Below.

Explanation:

Trigonometric Ratios are of 6 kinds.

$\sin x = \text{opposite"/"hypotenuse" = "perpendicular"/"hypotenuse}$

$\cos x = \text{adjacent"/"hypotenuse" = "base"/"hypotenuse}$

$\tan x = \text{opposite"/"adjacent" = "perpendicular"/"base}$

$\csc x = \text{hypotenuse"/"opposite" = "hypotenuse"/"perpendicular}$

$\sec x = \text{hypotenuse"/"adjacent" = "hypotenuse"/"base}$

$\cot x = \text{adjacent"/"opposite" = "base"/"perpendicular}$

So, you can see that, $\csc x$ is the inverse of $\sin x$.

So,

L.H.S = $\sin \frac{x}{\csc} x = \sin x \cdot \frac{1}{\csc} x = \sin x \cdot \sin x = {\sin}^{2} x$ = R.H.S

Hence Proved.