If for some alpha, beta such that alpha leq b | vedclass

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(B) We are given the equation $\sec^2(	an^{-1} \alpha) + \operatorname{cosec}^2(\cot^{-1} \beta) = 36$. Using the trigonometric identities $\sec^2(

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$14$

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(B) We are given the equation $\sec^2(	an^{-1} \alpha) + \operatorname{cosec}^2(\cot^{-1} \beta) = 36$. Using the trigonometric identities $\sec^2(	an^{-1} x) = 1 + 	an^2(	an^{-1} x) = 1 + x^2$ and $\operatorname{cosec}^2(\cot^{-1} x) = 1 + \cot^2(\cot^{-1} x) = 1 + x^2$,we substitute these into the equation: $(1 + \alpha^2) + (1 + \beta^2) = 36$ $\alpha^2 + \beta^2 = 34$. We are also given $\alpha + \beta = 8$. Squaring the sum: $(\alpha + \beta)^2 = 8^2 = 64$. $\alpha^2 + \beta^2 + 2\alpha\beta = 64$. Substituting $\alpha^2 + \beta^2 = 34$: $34 + 2\alpha\beta = 64 2\alpha\beta = 30 \alpha\beta = 15$. Now,we solve for $\alpha$ and $\beta$ using the sum and product: $x^2 - 8x + 15 = 0 (x - 3)(x - 5) = 0$. Since $\alpha \leq \beta$,we have $\alpha = 3$ and $\beta = 5$. Finally,$\alpha^2 + \beta = 3^2 + 5 = 9 + 5 = 14$.

List $I$	List $II$
$P$. $\left(\frac{1}{y^2}\left(\frac{\cos (\tan ^{-1} y)+y \sin (\tan ^{-1} y)}{\cot (\sin ^{-1} y)+\tan (\sin ^{-1} y)}\right)^2+y^4\right)^{1 / 2}$ takes value	$1$. $\frac{1}{2} \sqrt{\frac{5}{3}}$
$Q$. If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then possible value of $\cos \frac{x-y}{2}$ is	$2$. $\sqrt{2}$
$R$. If $\cos (\frac{\pi}{4}-x) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+\cos (\frac{\pi}{4}+x) \cos 2 x$ then possible value of $\sec x$ is	$3$. $\frac{1}{2}$
$S$. If $\cot (\sin ^{-1} \sqrt{1-x^2})=\sin (\tan ^{-1}(x \sqrt{6})), x \neq 0$, then possible value of $x$ is	$4$. $1$

If for some $\alpha, \beta$ such that $\alpha \leq \beta$ and $\alpha+\beta=8$,the equation $\sec^2(\tan^{-1} \alpha) + \operatorname{cosec}^2(\cot^{-1} \beta) = 36$ holds,then the value of $\alpha^2+\beta$ is . . . . . .

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