Q 129 - Precalculus Enhanced with Graphing Utilities

Question

Prove the reciprocal identities given in formula (2). 

Step-by-Step Solution

Answer

By using the trigonometry ratios in right angled triangle, we proved the following reciprocal identities :-

(a) $csc\theta =\frac{1}{\sin \theta }$

(b) $sec\theta =\frac{1}{\cos \theta }$

(c) $cot\theta =\frac{1}{\tan \theta }$

Step 1. Given Information

Here we have to prove the following reciprocal identities :-

(a) $csc\theta =\frac{1}{\sin \theta }$

(b) $sec\theta =\frac{1}{\cos \theta }$

(c) $cot\theta =\frac{1}{\tan \theta }$

We will apply trigonometry ratios, to prove these reciprocal identities.

Step 2. To prove identity (a) c s c θ = 1 sin θ .

Consider the following right angled triangle.

For this right angled triangle, sine function is defined as perpendicular upon hypotenuse.

That is :-

$\sin \theta =\frac{AB}{AC}$   $.......\left(1\right)$

and cosecant function is defined as hypotenuse upon perpendicular.

That is :-

$csc\theta =\frac{AC}{AB}$   $..........\left(2\right)$

From $\left(1\right)$, we have :-

$\sin \theta =\frac{AB}{AC}$

Take reciprocals on both sides, then we have :-

$\frac{1}{\sin \theta }=\frac{AC}{AB}$

By comparing $\left(2\right) and \left(3\right)$, we have :-

The right hand sides of both equations are equal, so left hand sides are also equal. This gives us :-

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

Hence proved.

Step 3. To prove identity (b) s e c θ = 1 cos θ

Consider the following right angled triangle.

For this right angled triangle, cosine function is defined as base upon hypotenuse.

That is :-

$\cos \theta =\frac{BC}{AC}$ $..........\left(4\right)$

and secant function is defined as hypotenuse upon base.

That is :-

$sec\theta =\frac{AC}{BC}$ $..........\left(5\right)$

From $\left(4\right)$, we have :- 

$\cos \theta =\frac{BC}{AC}$

Take reciprocals on both sides, then we have :- 

$\frac{1}{\cos \theta }=\frac{AC}{BC}$ $.........\left(6\right)$

By comparing $\left(5\right) and \left(6\right)$, we have :-

The right hand sides of both equations are equal, so left hand sides are also equal. This gives us :-

$sec\theta =\frac{1}{\cos \theta }$

Hence proved. 

Step 4. To prove identity (c) c o t θ = 1 tan θ

Consider the following right angled triangle.

For this right angled triangle, tangent function is defined as perpendicular upon base.

That is :-

$\tan \theta =\frac{AB}{BC}$ $........\left(7\right)$

and cotangent function is defined as base upon perpendicular. 

That is :-

$cot\theta =\frac{BC}{AB}$  $.........\left(8\right)$

From $\left(7\right)$, we have :-

$\tan \theta =\frac{AB}{BC}$

Take reciprocals on both sides, then we have :- 

$\frac{1}{\tan \theta }=\frac{BC}{AB}$ $.........\left(9\right)$

By comparing $\left(8\right) and \left(9\right)$, we have :-

The right hand sides of both equations are equal, so left hand sides are also equal. This gives us :-

$cot\theta =\frac{1}{\tan \theta }$

Hence proved.