If $\tan \theta=-\frac{4}{8}$, then $\sin \theta$ is equal to

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Let $f(\theta)=\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where $-\frac{\pi}{4}<\theta<\frac{\pi}{4}$. Then the value of $\frac{d}{d(\tan \theta)}(f(\theta))$ is

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If $\tan ^{-1} x+\tan ^{-1} y=\frac{\pi}{4}, x y<1$, then write the value of $x+y+x y$.

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If $x=\sin ^{-1}\left(3 t-4 t^3\right)$ and $y=\cos ^{-1}\left(\sqrt{1-t^2}\right)$, then $\frac{d y}{d x}$ is equal to

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If $A+B=225^{\circ}$, then $\frac{\cot A}{1+\cot A} \cdot \frac{\cot B}{1+\cot B}$ is equal to

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If $g$ is the inverse of a function $f$ and $f^{\prime}(x)=\frac{1}{1+x^5}$, then $g^{\prime}(x)$ equal to:

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Let $f:(-1,1) \rightarrow \operatorname{IR}$ be such that $f(\cos 4 \theta)=\frac{2}{2-\sec ^2 \theta}$ for $\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value $(s)$ of $f\left(\frac{1}{3}\right)$ is (are)

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The general value of $\theta$ satisfying the equation $2 \sin ^2 \theta-3 \sin \theta-2=0$ is

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यदि $\tan \theta=\frac{8}{15}$ हो, तो $\operatorname{cosec} \theta$ का मान है

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Prove that : $\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{\prime}\left(\frac{1}{5}\right)+\tan ^{\prime}\left(\frac{1}{8}\right)=\frac{\pi}{4}$

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Prove the following : $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{8}\right)=\frac{\pi}{4}$.

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The value of $\frac{\cos 12^{\circ}-\sin 12^{\circ}}{\cos 12^{\circ}+\sin 12^{\circ}}+\frac{\sin 147^{\circ}}{\cos 147^{\circ}}$ is equal to

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If $\tan \theta=-\frac{4}{8}$, then $\sin \theta$ is equal to

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