Trigonometry

If $\cos\alpha+\cos\beta=a$, $\sin\alpha+\sin\beta=b$ and $\theta$ is the arithmetic mean between $\alpha$ and $\beta$, then $\sin2\theta+\cos2\theta$ is equal to

The maximum value of $(\cos\alpha_1)(\cos\alpha_2)\cdots(\cos\alpha_n)$ where $0\le \alpha_1,\alpha_2,\ldots,\alpha_n\le\pi$ and $(\cot\alpha_1)(\cot\alpha_2)\cdots(\cot\alpha_n)=1$ is

If $y=mx$ bisects the angle between the lines $x^2(\tan^2\theta+\cos^2\theta)+2xy\tan\theta-y^2\sin\theta=0$ when $\theta=\dfrac{\pi}{3}$, then the value of $\sqrt{3}m^2+4m$ is

If $y=\sec^{-1}\left(\frac{x+1}{x-1}\right)+\sin^{-1}\left(\frac{x-1}{x+1}\right)$, $x\in[0,\infty)$ and $x\ne1$, then $\dfrac{dy}{dx}$ is equal to

The function $f(x)=2\sin x+\sin 2x,\ x\in[0,2\pi]$ has absolute maximum and minimum at

The value of $\displaystyle \int_0^{\pi/2} \frac{dx}{1+\tan^3 x}$ is

If $A = \cos^2\theta + \sin^4\theta$, then for all values of $\theta$:

If $ \theta = \tan^{-1}\dfrac{1}{1+2} + \tan^{-1}\dfrac{1}{1+2\cdot3} + \tan^{-1}\dfrac{1}{1+3\cdot4} + \ldots + \tan^{-1}\dfrac{1}{1+n(n+1)} $, then $\tan\theta$ is equal to:

$ \displaystyle \text{The value of } \frac{1 - \tan^{2} 15^\circ}{1 + \tan^{2} 15^\circ} \text{ is:} $

The general solution of $ \sqrt{3}\cos x + \sin x = 3 $ is: