If $P = \frac{1}{2} \sin^2 \theta + \frac{1}{3} \cos^2 \theta$, then:

The maximum value of the function $y = e^{5 + \sqrt{3} \sin x + \cos x}$ is

The extremum values of the function $f(x) = \frac{1}{\sin x + 4} - \frac{1}{\cos x - 4}$ for $x \in R$ are:

$a, b, c$ are the sides of a scalene triangle $ABC$. If angles $\alpha, \beta, \gamma$ lie between $0$ and $\pi$ such that $\cos \alpha = \frac{a}{b+c}, \cos \beta = \frac{b}{c+a}$ and $\cos \gamma = \frac{c}{a+b}$,then $\tan^2 \frac{\alpha}{2} + \tan^2 \frac{\beta}{2} + \tan^2 \frac{\gamma}{2} =$

If $P+Q+R=\frac{\pi}{4}$,then $\cos \left(\frac{\pi}{8}-P\right)+\cos \left(\frac{\pi}{8}-Q\right)+\cos \left(\frac{\pi}{8}-R\right)=$

The equation $\sin x(\sin x+\cos x)=k$ has real solutions, where $k$ is a real number. Then,