Chapter 7: Q. 90 (page 487)

Write the trigonometric expression as an algebraic expression containing uand v
$s e c (\tan^{- 1} u + \cos^{- 1} v)$

Short Answer

Expert verified

$= \frac{1}{\sqrt{1 + u^{2}} (v - u \sqrt{1 - v^{2}})}$

Step by step solution

Step 1. Given information

a trigonometric equation $s e c (\tan^{- 1} u + \cos^{- 1} v)$

Step 2. Putting tan-1u=xu=tanxcos-1v=yv=cosy

We know that,

$s e c x = \sqrt{1 + \tan^{2} x}$

also, $s e c x = \sqrt{1 + u^{2}}$

using the reciprocal identity of secx we get

$\cos x = \sqrt{1 + u^{2}}$

Step 3. Using identity,

$\tan x = \frac{\sin x}{\cos x}$

$\sin x = u \sqrt{1 + u^{2}}$

Also,

$\sin y = \sqrt{1 - v^{2}}$

Step 4. Using the formula cosx+y=cosxcosy-sinxsiny

$\frac{1}{\cos (x + y)} = \frac{1}{v \sqrt{1 + u^{2}} - (u \sqrt{1 + u^{2}}) (\sqrt{1 - v^{2}})} \frac{1}{\cos (x + y)} = \frac{1}{\sqrt{1 + u^{2}} (v - u \sqrt{1 - v^{2}})}$

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Write the trigonometric expression as an algebraic expression containing uand v
$s e c (\tan^{- 1} u + \cos^{- 1} v)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Putting tan-1u=xu=tanxcos-1v=yv=cosy

Step 3. Using identity,

Step 4. Using the formula cosx+y=cosxcosy-sinxsiny

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Write the trigonometric expression as an algebraic expression containing uand vsec(tan-1u+cos-1v)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Putting tan-1u=xu=tanxcos-1v=yv=cosy

Step 3. Using identity,

Step 4. Using the formula cosx+y=cosxcosy-sinxsiny

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Write the trigonometric expression as an algebraic expression containing uand v
$s e c (\tan^{- 1} u + \cos^{- 1} v)$