If \(\cot ^{-1}(\sqrt{\sin \alpha})+\tan ^{-1}(\sqrt{\sin \alpha})=u\), then cos2u is equal to

(a)

tan²\(\alpha\)

(b)

0

(c)

-1

(d)

tan2\(\alpha\)

If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to

(a)

tan^-1x

(b)

sin^-1x

(c)

0

(d)

\(\pi\)

If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is

(a)

4

(b)

5

(c)

2

(d)

3

sin (tan^-1x), |x| < 1 is equal to

(a)

\(\frac{x}{\sqrt{1-x^2}}\)

(b)

\(\frac{1}{\sqrt{1-x^2}}\)

(c)

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

(d)

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

The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

(a)

2

(b)

3

(c)

1

(d)

none

\(sin\left\{ 2{ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \right\} =\) __________

(a)

\(\frac { 6 }{ 15 } \)

(b)

\(\frac { 24 }{ 25 } \)

(c)

\(\frac { 4 }{ 5 } \)

(d)

\(\frac { -24 }{ 25 } \)

The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

(a)

\(\frac { 19 }{ 8 } \)

(b)

\(\frac { 8 }{ 19 } \)

(c)

\(\frac { 19 }{ 12 } \)

(d)

\(\frac { 3 }{ 4 } \)

(1) cot(cot^-1(+600)) = -600
(2) cot(cot^-1(1782)) = 1782
(3) \(cot\left( { cot }^{ -1 }\left( \frac { -17 }{ 9 } \right) \right) =\frac { -17 }{ 9 } \)
(4) \(cot({ cot }^{ -1 }\left( \sqrt { 3 } \right) =\sqrt { 3 } \)

Find the principal value of
 \({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

Find all the values of x such that -10\(\pi\)\(\le x\le\)10\(\pi\) and sin x = 0 

Find the period and amplitude of y = 4sin(−2x)

Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)

If \({ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) ={ tan }^{ -1 }x\) then find the value of x,

Find tan(tan^-1(2019))

Solve: cos(tan^-1x) = \(sin\left( { cot }^{ -1 }\frac { 3 }{ 4 } \right) \) 

Find cos^-1 \((-\frac{1}{\sqrt2})\)

Find the principal value of
sec⁻¹(−2).

If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x² = sin 2a