If $f(x) = a \cos(bx + c) + d$,then the range of $f(x)$ is

The maximum value of $\left(2 \cos^2 18^{\circ} - \sin 18^{\circ}\right) \left(\cos \theta + 3 \sqrt{2} \cos \left(\theta + \frac{\pi}{4}\right) + 3\right)$ is

The maximum and minimum values of the function $|\sin 4x + 3|$ are

If $ABC$ is not a right-angled triangle and $\sin \left(\frac{\pi}{4}-A\right) \sin \left(\frac{\pi}{4}-B\right) = -\frac{1}{2 \sqrt{2}} \operatorname{cosec}\left(\frac{\pi}{4}-C\right)$,then $\tan A \tan B + \tan B \tan C + \tan C \tan A = $

Let $x, y$ be positive real numbers and $m, n$ be positive integers. The maximum value of the expression $\frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})}$ is

The value of $x$ for the maximum value of $(\sqrt{3} \sin x + \cos x)$ is .....$^o$.