If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1 x + cos^-1 y is equal to

(a)

\(\frac{2\pi}{3}\)

(b)

\(\frac{\pi}{3}\)

(c)

\(\frac{\pi}{6}\)

(d)

\(\pi\)

If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

(a)

0

(b)

1

(c)

2

(d)

3

The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

(a)

[1, 2]

(b)

[-1, 1]

(c)

[0, 1]

(d)

[-1, 0]

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to

(a)

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

(b)

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

(c)

\(\frac { 1 }{ 2 } {tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

(d)

\({ tan}^{ -1 }\left( \frac { 1}{ 2 } \right) \)

sin^-1(2cos²x-1)+cos^-1(1-2sin²x)=

(a)

\(\frac{\pi}{2}\)

(b)

\(\frac{\pi}{3}\)

(c)

\(\frac{\pi}{4}\)

(d)

\(\frac{\pi}{6}\)

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)

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

If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________

(a)

\(\frac { 1 }{ 2 } \)

(b)

\(\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(\frac { -1 }{ 2 } \)

(d)

none of these

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

(a)

\(\frac { \pi }{ 6 } \)

(b)

\(\frac { \pi }{ 3 } \)

(c)

\(\frac { \pi }{ 2 } \)

(d)

\(\frac { -\pi }{ 3 } \)

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

(a)

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

(b)

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

(c)

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

(d)

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

In a \(\Delta ABC\)  if C is a right angle, then  \({ tan }^{ -1 }\left( \frac { a }{ b+c } \right) +{ tan }^{ -1 }\left( \frac { b }{ c+a } \right) =\) ________

(a)

\(\frac { \pi }{ 3 } \)

(b)

\(\frac { \pi }{ 4 } \)

(c)

\(\frac { 5\pi }{ 2 } \)

(d)

\(\frac { \pi }{ 6 } \)