The total number of real solutions of the equation

$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1}\left(\frac{6 \tan \theta}{9 + \tan^2 \theta}\right) $

is

(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\tan^{-1} x$ assume values in $[ -\frac{\pi}{2}, \frac{\pi}{2}]$ and $( -\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)

Considering only the principal values of the inverse trigonometric functions, the value of

$ \tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) $

is

$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ List-$I$
(P.)$\,\,\,\,$ Let $y\left( x \right) = \cos \left( {3{{\cos }^{ - 1}}x} \right),x \in \left[ { - 1,1} \right],x \ne \pm {{\sqrt 3 } \over 2}.$ Then ${1 \over {y\left( x \right)}}\left\{ {\left( {{x^2} - 1} \right){{{d^2}y\left( x \right)} \over {d{x^2}}} + x{{dy\left( x \right)} \over {dx}}} \right\}$ equals
(Q.)$\,\,\,\,$ Let ${A_1},{A_2},....,{A_n}\left( {n > 2} \right)$ be the vertices of a regular polygon of $n$ sides with its centre at the origin. Let ${\overrightarrow {{a_k}} }$ be the position vector of the point ${A_k},k = 1,2,......,n.$ $$f\left| {\sum\nolimits_{k = 1}^{n - 1} {\left( {\overrightarrow {{a_k}} \times \overrightarrow {{a_{k + 1}}} } \right)} } \right| = \left| {\sum\limits_{k = 1}^{n - 1} {\left( {\overrightarrow {{a_k}} .\,\overrightarrow {{a_{k + 1}}} } \right)} } \right|,$$ then the minimum value of $n$ is
(R.)$\,\,\,\,$ If the normal from the point $P(h, 1)$ on the ellipse ${{{x^2}} \over 6} + {{{y^2}} \over 3} = 1$ is perpendicular to the line $x+y=8,$ then the value of $h$ is
(S.)$\,\,\,\,$ Number of positive solutions satisfying the equation ${\tan ^{ - 1}}\left( {{1 \over {2x + 1}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {4x + 1}}} \right) = {\tan ^{ - 1}}\left( {{2 \over {{x^2}}}} \right)$ is

$\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$List-$II$
(1.)$\,\,\,\,$ $1$
(2.)$\,\,\,\,$ $2$
(3.)$\,\,\,\,$ $8$
(4.)$\,\,\,\,$ $9$

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 $

$3.$ $\,\,\,\,\,$ ${1 \over 2}$

$1.$ $\,\,\,\,$ $1$

Let $(x,y)$ be such that ${\sin ^{ - 1}}(ax) + {\cos ^{ - 1}}(y) + {\cos ^{ - 1}}(bxy) = {\pi \over 2}$.

Match the statements in Column I with the statements in Column II.

Let F(x) be an indefinite integral of $\sin^2x$.

Statement 1 : The function F(x) satisfies F($x+\pi$) = F($x$) for all real x.

Statement 2 : ${\sin ^2}(x + \pi ) = {\sin ^2}x$ for all real x.