# Trigonometric reciprocal relationships

The trigonometric reciprocal relationships is the simplest and most basic trigonometric identities.
Reciprocals of a number means 1 divided by that number
For example, the reciprocal of 4 is $\frac{1}{4}$
In case of trigonometry, the reciprocal of $sin \Theta = \frac{1}{csc \Theta}$ and so on.
In this section we will prove the trigonometric reciprocal identities.
1) Prove that :
A) $sin \Theta = \frac{1}{csc \Theta}$ OR $csc \Theta = \frac{1}{sin \Theta}$

B) $cos \Theta = \frac{1}{sec \Theta}$ OR $sec \Theta = \frac{1}{cos \Theta}$

C)$tan \Theta = \frac{1}{cot \Theta}$ OR $cot \Theta = \frac{1}{tan \Theta}$

Proof : Let OA be the ray which makes an angle $\Theta$ with the x-axis. This angle $\Theta$ in any of the four quadrants. From point 'P' on the ray OA ,draw a perpendicular to ray OB. So PQ $\perp$ OB. In triangle POQ, we have
$OP^{2} = OQ^{2} + PQ^{2}$
(A)
$sin \Theta = \frac{opposite}{hypotenuse} = \frac{PQ}{OP}$ -----(i)

$csc \Theta = \frac{Hypotenuse}{opposite} = \frac{OP}{PQ}$ -----(ii)

∴ from (i)
$sin \Theta = \frac{1}{\frac{OP}{PQ}}$ ⇒ $sin \Theta = \frac{1}{csc \Theta}$

Now, from(ii), $csc \Theta = \frac{1}{\frac{PQ}{OP}}$ ⇒ $csc \Theta = \frac{1}{sin \Theta}$

So, $sin \Theta$ and $csc \Theta$ are reciprocal of each other.

(B)
$cos \Theta = \frac{Adjacent}{hypotenuse} = \frac{OQ}{OP}$ -----(iii)

$sec \Theta = \frac{Hypotenuse}{adjacent} = \frac{OP}{OQ}$ -----(iv)

∴ from (iii)
$cos \Theta = \frac{1}{\frac{OP}{OQ}}$ ⇒ $cos \Theta = \frac{1}{sec \Theta}$

Now, from(iv),
$sec \Theta = \frac{1}{\frac{OQ}{OP}}$ ⇒ $sec \Theta = \frac{1}{cos \Theta}$

So, $cos \Theta$ and $sec \Theta$ are reciprocal of each other.

(C)
$tan \Theta = \frac{opposite}{Adjacent} = \frac{PQ}{OQ}$ -----(v)

$cot \Theta = \frac{Adjacent}{opposite} = \frac{OQ}{PQ}$ -----(vi)

∴ from (v)
$tan \Theta = \frac{1}{\frac{OQ}{PQ}}$ ⇒ $tan \Theta = \frac{1}{cot \Theta}$

Now, from(vi),
$cot \Theta = \frac{1}{\frac{PQ}{OQ}}$ ⇒ $cot \Theta = \frac{1}{tan \Theta}$

So, $tan \Theta$ and $cot \Theta$ are reciprocal of each other.

## Trigonometric reciprocal relationships

1) $sin \Theta \times csc \Theta$ = 1
2) $cos \Theta \times sec \Theta$ = 1
3) $tan \Theta \times cot \Theta$ = 1