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Basic Inverse Trigonometric Functions

Identify the ...

Question

Identify the pair of functions which are not identical

A

y = t a n (c o s^{- 1} x); y = \frac{\sqrt{1 - x^{2}}}{x}

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B

y = t a n (c o t^{- 1} x); y = \frac{1}{x}

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C

y = s i n (a r c t a n x); y = \frac{x}{\sqrt{1 + x^{2}}}

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D

y = c o s (t a n^{- 1} x); y = s i n (t a n^{- 1} x)

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Solution

The correct option is D $y = c o s (t a n^{- 1} x); y = s i n (t a n^{- 1} x)$
Conceptual, verify each of options

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Similar questions

Q. Identify the pair(s) of functions which are identical.
i)

y = sin (a r c tan x)

y = \frac{x}{\sqrt{1 + x^{2}}}

y = cos (a r c tan x)

y = sin (a r c cot x)

y = tan ({cos}^{- 1} x)

y = \frac{\sqrt{1 - x^{2}}}{x}

y = tan ({cot}^{- 1} x)

y = \frac{1}{x}

Inverse circular functions,Principal values of

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

solve for x the following equations :

{c o t}^{- 1} x + {s i n}^{- 1} \frac{1}{\sqrt (5)} = \frac{π}{4}

2 {t a n}^{- 1} (c o s x) = t a n {t a n}^{- 1} (2 c o s e c x)

t a n ({c o s}^{- 1} x) = s i n ({c o t}^{- 1} \frac{1}{2})

Q. Inverse circular functions,Principal values of

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

.

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

s i n ({c o s}^{- 1} \frac{3}{5})

c o s ({t a n}^{- 1} \frac{3}{4})

c o s ({t a n}^{- 1} x)

s i n ({c o t}^{- 1} (1 + x)) = c o s ({t a n}^{- 1} x)

1 / 2

1

0

- 1 / 2

s i n ({c o t}^{- 1} x)

s i n (2 {s i n}^{- 1} 0.8)

Q. Inverse circular functions,Principal values of

{sin}^{- 1} x, {cos}^{- 1} x, {tan}^{- 1} x

.

{tan}^{- 1} x + {tan}^{- 1} y = {tan}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {tan}^{- 1} \frac{x + y}{1 - x y}

x y > 1

{tan}^{- 1} \frac{1 - x}{1 + x} {tan}^{- 1} \frac{1 - y}{1 + y} = {sin}^{- 1} \frac{y - x}{\sqrt{1 + x^{2}} \sqrt{1 + y^{2}}}

Q. Inverse circular functions,Principal values of

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

.

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

0 < x < 1

\sqrt{1 + x^{2}} {[{x c o s ({c o t}^{- 1} x) + s i n ({c o t}^{- 1} x)}^{2} - 1]}^{1 / 2}

is equal to
(a)

\frac{x}{\sqrt{1 + x^{2}}}

x

x \sqrt{1 + x^{2}}

\sqrt{1 + x^{2}}

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