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If the equations $x^{3} + b x^{2} + c x + 1 = 0, (b < c)$ has only one real root $α$ . Then the value of $2 \tan^{- 1} (cosec α) + \tan^{- 1} (2 \sin α \sec^{2} α)$ ,is

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Important Questions on Inverse Trigonometric Functions

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

Considering only the principal values of inverse functions, the set

A = \{x \geq 0 : \tan^{- 1} (2 x) + \tan^{- 1} (3 x) = \frac{π}{4}\}

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

S

is the sum of the first 10 terms of the series,

\tan^{- 1} (\frac{1}{3}) + \tan^{- 1} (\frac{1}{7}) + \tan^{- 1} (\frac{1}{13}) + \tan^{- 1} (\frac{1}{21}) + \dots \dots

then

\tan (S)

is equal to :

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

The value of

S = \sum_{n = 1}^{\infty} \tan^{- 1} \frac{2 n}{n^{4} + n^{2} + 2}

is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

\sum_{k = 1}^{n} \tan^{- 1} (\frac{1}{k^{2} + k + 1}) = \tan^{- 1} (θ)

, then

θ =

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

\sin^{- 1} (\frac{x}{5}) + {c o s e c}^{- 1} (\frac{5}{4}) = \frac{π}{2},

then the value of

x

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

2 \tan^{- 1} \frac{1}{5} + \sec^{- 1} \frac{5 \sqrt{2}}{7} + 2 \tan^{- 1} \frac{1}{8} =

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

2 π - (\sin^{- 1} \frac{4}{5} + \sin^{- 1} \frac{5}{13} + \sin^{- 1} \frac{16}{65})

is equal to :

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

The value of

\sin (2 \sin^{- 1} 0.8)

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

The value of

4 \tan^{- 1} (\frac{1}{5}) - \frac{π}{4} =

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

α = {c o s}^{- 1} (\frac{3}{5})

β = {t a n}^{- 1} (\frac{1}{3})

, where

0 < α, β < \frac{π}{2},

then

α - β

is equal to

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

\cos^{- 1} (\frac{x}{2}) + \cos^{- 1} (\frac{y}{3}) = θ,

then

9 x^{2} - 12 x y \cos θ + 4 y^{2} =

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

For any positive integer

n

, define

f_{n} : (0, \infty) \to R

f_{n} (x) = \sum_{i = 1}^{n} \tan^{- 1} (\frac{1}{1 + (x + j) (x + j - 1)})

for all

x \in (0, \infty)

$($ Here, the inverse trigonometric function $\tan^{- 1} x$ assumes values in $(- \frac{π}{2}, \frac{π}{2})$ $)$

Then, which of the following statement(s) is (are) TRUE?

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

0 \leq A \leq \frac{π}{4}

, then

\tan^{- 1} (\frac{1}{2} \tan 2 A) + \tan^{- 1} (\cot A) + \tan^{- 1} (\cot^{3} A)

is equal to

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

\sum_{m = 1}^{n} \tan^{- 1} (\frac{2 m}{m^{4} + m^{2} + 2})

is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

\sin^{- 1} \frac{8}{17} + \sin^{- 1} \frac{3}{5}

is equal to

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

\cot^{- 1} \frac{1}{5} + \cot^{- 1} \frac{1}{3} - \cot^{- 1} \frac{4}{7} = \cot^{- 1} x

, then

x

is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

The value of the expression

\tan [(1 / 2) \cos^{- 1} (2 / \sqrt{5})]

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

The value of

\sin^{- 1} (\frac{12}{13}) - \sin^{- 1} (\frac{3}{5})

is equal to:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

f (x) = 2 \tan^{- 1} x + \sin^{- 1} (\frac{2 x}{1 + x^{2}}), x > 1

, then

f (5)

is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Identities of Inverse Trigonometric Functions

Let

\tan^{- 1} y = \tan^{- 1} x + \tan^{- 1} (\frac{2 x}{1 - x^{2}})

, where

|x| < \frac{1}{\sqrt{3}}

,Then a value of

y

If the equations x3+bx2+cx+1=0,(b<c) has only one real root α. Then the value of 2tan-1(cosecα)+tan-12sinαsec2α,is

Important Questions on Inverse Trigonometric Functions

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If the equations $x^{3} + b x^{2} + c x + 1 = 0, (b < c)$ has only one real root $α$ . Then the value of $2 \tan^{- 1} (cosec α) + \tan^{- 1} (2 \sin α \sec^{2} α)$ ,is