Chapter 7: Q. 77 (page 475)

Establish each identity.
$\frac{{(s e c v - \tan v)}^{2} + 1}{c s c v (s e c v - \tan v)} = 2 \tan v$

Short Answer

Expert verified

The left side of the given equation has been simplified to bring the right side.

Step by step solution

Step 1. Given Information

The given equation is $\frac{{(s e c v - \tan v)}^{2} + 1}{c s c v (s e c v - \tan v)} = 2 \tan v$

Step 2. Calculation

Consider the left side of the given equation and simplify to bring the right side in order to establish the equation.

$\frac{{(s e c v - \tan v)}^{2} + 1}{c s c v (s e c v - \tan v)} = \frac{{(\frac{1}{\cos v} - \frac{\sin v}{\cos v})}^{2} + 1}{\frac{1}{\sin v} (\frac{1}{\cos v} - \frac{\sin v}{\cos v})} = \frac{\frac{{(1 - \sin v)}^{2}}{\cos^{2} v} + 1}{\frac{1}{\sin v} (\frac{1 - \sin v}{\cos v})} = \frac{\frac{{(1 - \sin v)}^{2} + \cos^{2} v}{\cos^{2} v}}{\frac{1 - \sin v}{\sin v \cos v}} = \frac{1 + \sin^{2} v + \cos^{2} v - 2 \sin v}{\cos v} \times \frac{\sin v}{1 - \sin v} = \frac{2 - 2 \sin v}{1 - \sin v} \times \frac{\sin v}{\cos v} = \frac{2 (1 - \sin v)}{1 - \sin v} \times \tan v = 2 \tan v$

Hence, Proved.

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Establish each identity.
$\frac{{(s e c v - \tan v)}^{2} + 1}{c s c v (s e c v - \tan v)} = 2 \tan v$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Establish each identity.(secv-tanv)2+1cscv(secv-tanv)=2tanv

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Step by step solution

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Most popular questions from this chapter

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Establish each identity.
$\frac{{(s e c v - \tan v)}^{2} + 1}{c s c v (s e c v - \tan v)} = 2 \tan v$