List $$I$$
$$P.$$$$\,\,\,\,\,$$ $${\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }^{ - 1}}y} \right)} \over {\cot \left( {{{\sin }^{ - 1}}y} \right) + \tan \left( {{{\sin }^{ - 1}}y} \right)}}} \right)}^2} + {y^4}} \right)^{1/2}}$$ takes value
$$Q.$$ $$\,\,\,\,$$ If $$\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z$$ then
possible value of $$\cos {{x - y} \over 2}$$ is
$$R.$$ $$\,\,\,\,\,$$ If $$\cos \left( {{\pi \over 4} - x} \right)\cos 2x + \sin x\sin 2\sec x = \cos x\sin 2x\sec x + $$
$$\cos \left( {{\pi \over 4} + x} \right)\cos 2x$$ then possible value of $$\sec x$$ is
$$S.$$ $$\,\,\,\,\,$$ If $$\cot \left( {{{\sin }^{ - 1}}\sqrt {1 - {x^2}} } \right) = \sin \left( {{{\tan }^{ - 1}}\left( {x\sqrt 6 } \right)} \right),\,\,x \ne 0,$$
Then possible value of $$x$$ is
List $$II$$
$$1.$$ $$\,\,\,\,\,$$ $${1 \over 2}\sqrt {{5 \over 3}} $$
$$2.$$ $$\,\,\,\,\,$$ $$\sqrt 2 $$
$$3.$$ $$\,\,\,\,\,$$ $${1 \over 2}$$
$$1.$$ $$\,\,\,\,$$ $$1$$
$$\cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} } \right)$$
$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {1 + 2 \times {{n(n + 1)} \over 2}} \right)} } \right)$$
$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}(1 + n(n + 1))} } \right)$$
$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}\left( {{{n(n + 1) + 1} \over {(n + 1) - n}}} \right)} } \right)$$
$$ = \cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}n - {{\cot }^{ - 1}}(n + 1)} } \right)$$
$$ = \cot (({\cot ^{ - 1}}1 + {\cot ^{ - 1}}2 + {\cot ^{ - 1}}3 + .... + {\cot ^{ - 1}}23) - ({\cot ^{ - 1}}2 + {\cot ^{ - 1}}3 + .... + {\cot ^{ - 1}}23 + {\cot ^{ - 1}}24))$$
$$ = \cot ({\cot ^{ - 1}}1 - {\cot ^{ - 1}}24)$$
$$ = \cot \left( {{{\cot }^{ - 1}}{{24 \times 1 + 1} \over {24 - 1}}} \right) = {{25} \over {23}}$$