One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval

Let $\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots$ up to $40$ terms. If $(\tan \beta)^{1020}$ is a root of the equation $x^2 + x - 2 = 0$, where $\beta \in (0, \frac{\pi}{2})$, then $\sin^2 \beta + 3 \cos^2 \beta$ is equal to:

Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$,where $x \in R$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ is equal to

If $0 \leq x \leq \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$ is

If $3 \sin (\alpha-\beta)=5 \cos (\alpha+\beta)$ and $\alpha+\beta \neq \frac{\pi}{2}$,then $\frac{\tan \left(\frac{\pi}{4}-\alpha\right)}{\tan \left(\frac{\pi}{4}-\beta\right)}=$

If $\tan(\pi \sin \theta) = \cot(\pi \cos \theta)$,then $\left| \cot \left( \theta - \frac{\pi}{4} \right) \right|$ is -