Chapter 7: Q. 89 (page 487)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. The given information

trigonometric equation $\tan (\sin^{- 1} u - \cos^{- 1} v)$

Step 2. Putting sin-1u=xu=sinxcos-1v=yv=cosy

We know that,

$\cos x = \sqrt{1 - \sin^{2} x}$

now,

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

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

$= \frac{u}{\sqrt{1 - u^{2}}}$

Step 3. Similarly

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

Step 4. Using the formula tanx-y=tanx-tany1+tanxtany

Solving we get,

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

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

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

Short Answer

Step by step solution

Step 1. The given information

Step 2. Putting sin-1u=xu=sinxcos-1v=yv=cosy

Step 3. Similarly

Step 4. Using the formula tanx-y=tanx-tany1+tanxtany

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Write the trigonometric expression as an algebraic expression containing uand vtansin-1u-cos-1v

Short Answer

Step by step solution

Step 1. The given information

Step 2. Putting sin-1u=xu=sinxcos-1v=yv=cosy

Step 3. Similarly

Step 4. Using the formula tanx-y=tanx-tany1+tanxtany

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Statistics

Pure Maths

Decision Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

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