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Chapter 4: Problem 14

Indicate the relation which is true. (a) \(\tan \left|\tan ^{-1} x\right|=|x|\) (b) \(\cot \left|\cot ^{-1} x\right|=x\) (c) \(\tan ^{-1}|\tan x|=|x|\) (d) \(\sin \left|\sin ^{-1} x\right|=|x|\)

Short Answer

Expert verified
The correct relation is (c) \(\tan ^{-1}|\tan x|=|x|\), as this relation holds true for all \(x\). The other given relations do not hold true due to the properties of trigonometric functions and their inverses not preserving absolute values.

Step by step solution

01

Relation (a)

We will evaluate the relation \(\tan \left|\tan ^{-1} x\right|=|x|\): Let \(y = \tan^{-1} x\), then \(\tan y = x\). The given relation is now expressed as: \(\tan |y| = |x|\) Since the tangent function does not preserve the absolute value, \(\tan |y|\) does not equal \(|\tan y|\), which means the relation is not true.
02

Relation (b)

We will evaluate the relation \(\cot \left|\cot ^{-1} x\right|=x\): Let \(y = \cot^{-1} x\), then \(\cot y = x\). The given relation is now expressed as: \(\cot |y| = |x|\) Since the cotangent function does not preserve the absolute value, \(\cot |y|\) does not equal \(|\cot y|\), which means the relation is not true.
03

Relation (c)

We will evaluate the relation \(\tan ^{-1}|\tan x|=|x|\): Let \(y = \tan^{-1} |\tan x|\), then \(|x| = |\tan( \tan^{-1}x)|\). This gives us: \(|x| = |\tan(\alpha)| \Rightarrow x = \tan(\alpha)\), for some angle \(\alpha\). Then, the relation is true for all \(x\).
04

Relation (d)

We will evaluate the relation \(\sin \left|\sin ^{-1} x\right|=|x|\): Let \(y = \sin^{-1} x\), then \(\sin y = x\). The given relation is now expressed as: \(\sin |y| = |x|\) Since the sine function also does not preserve the absolute value, \(\sin |y|\) does not equal \(|\sin y|\), which means the relation is not true. By evaluating each of the given relations, we find that among the given options in this exercise, only option (c) is true.

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Most popular questions from this chapter

The interval on which \(\cos ^{-1} x>\sin ^{-1} x>\tan ^{-1} x\) is (a) \(\left(0, \frac{1}{\sqrt{2}}\right)\) (b) \([-1,1]\) (c) \((0,1]\) (d) None of these

The value of \(x\), if \(2 \sin ^{-1} x=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)\) is (a) \(\left[-\frac{1}{\sqrt{2}}, 1\right]\) (b) \(\left[-\frac{1}{\sqrt{2}}, 1\right]\) (c) \(\left[-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]\) (d) \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

\(\sin ^{-1}(-x)\) is equal to (a) \(-\sin ^{-1} x\) (b) \(\pi+\sin ^{-1} x\) (c) \(2 \pi-\sin ^{-1} x\) (d) \(3 \pi-\cos ^{-1} \sqrt{1-x^{2}}, x>0\)

If \(\tan ^{-1} y=4 \tan ^{-1} x\), then \(y\) is not finite if (a) \(x^{2}=3+2 \sqrt{2}\) (b) \(x^{2}=3-2 \sqrt{2}\) (c) \(x^{4}=6 x^{2}-1\) (d) \(x^{4}=6 x^{2}+1\)

If the solution set of system of equations \(3 \sin ^{-1} x=-\pi\) \(-\sin ^{-1}\left(3 x-4 x^{3}\right)\) is \([a, b]\) and that of \(3 \sin ^{-1} x=\pi-\sin ^{-1}\) \(\left(3 x-4 x^{3}\right)\) is \([\mathrm{c}, \mathrm{d}]\) then evaluate \(|a|+|b|+|c|+|d|\)
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