Trigonometric Identities for Product to Sum or Difference

Testing purpose - mathemetics

The value of the expression $\frac{2(\mathrm{s}\mathrm{i}\mathrm{n}1°+\mathrm{s}\mathrm{i}\mathrm{n}2°+\mathrm{s}\mathrm{i}\mathrm{n}3°+........+\mathrm{s}\mathrm{i}\mathrm{n}89°)}{2(\mathrm{c}\mathrm{o}\mathrm{s}1°+\mathrm{c}\mathrm{o}\mathrm{s}2°+\mathrm{c}\mathrm{o}\mathrm{s}3°+........+\mathrm{c}\mathrm{o}\mathrm{s}44°)+1}$ is equal to

The value of ${\cos }^2(\alpha +\beta +\gamma )+{\cos }^2(\beta +\gamma )+{\cos }^2\alpha -2\cos \alpha \cos (\beta +\gamma )\cos (\alpha +\beta +\gamma )$ is

Value of $2\sin 48°\sin 12°$ is :

If $\cos \left(x-y\right), \cos x$ and $\cos (x+y)$ are in $\mathrm{H}.\mathrm{P}.$, then the value of $\frac{5}{{\left|\cos x\sec \frac{y}{2}\right|}^2}$ is

If $\cos \theta \cos 2\theta \cos 3\theta =\frac{1}{4}(0\le \theta \le \pi )$ have 4 roots which are in A.P. with first term $a$ and common
difference $d$ then  $a+\mathrm{d}$ can be ? (given that $\cos 2\theta \ne -\frac{1}{2}$ )

If $\underset{0}{\overset{4\pi }{\int }}\ln |13\text{\hspace{0.17em}}\sin x+3\sqrt{3}\cos x|dx=k\pi \hspace{0.17em}\ln 7$, then the value of $k$ is

If $2\sin C\cos A=\sin B$$,$ then $\mathrm{\Delta }ABC$ is

Let $\mathrm{\alpha }\pm \mathrm{\beta }$ is not an odd multiple of $\mathrm{\pi }$ . If $\cos \mathrm{\alpha }+\cos \mathrm{\beta }=\mathrm{b},\sin \mathrm{\alpha }+\sin \mathrm{\beta }=\mathrm{a},\mathrm{ }\mathrm{\theta }=\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}$ and $\sin 2\mathrm{\theta }+\cos 2\mathrm{\theta }=1+\frac{\mathrm{n}\mathrm{a}\left(\mathrm{a}-\mathrm{b}\right)}{{\mathrm{a}}^2+{\mathrm{b}}^2}$ where $\mathrm{n}\mathrm{ }\mathrm{ϵ}\mathrm{ }\mathrm{I},$ then

If degree of polynomial obtained in above question is p then max. value of $(\mathrm{p}\mathrm{ }+\mathrm{ }1)\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{ }\mathrm{x}\mathrm{ }+\mathrm{ }$

(p + 2) cos x is-

If $\frac{\tan \left(\mathrm{\alpha }+\mathrm{\beta }-\mathrm{\gamma }\right)}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathrm{\alpha }-\mathrm{\beta }+\mathrm{\gamma }\right)}=\frac{\tan \gamma }{\tan \beta }\left(\beta \ne \gamma , \beta \ne n\pi , \alpha -\beta +\gamma \ne n\pi , n\in I,\beta -\gamma \ne m\pi , m\in I\right)$ then  $\left(\sin 2\alpha +\sin 2\beta +\sin 2\gamma \right)$ is equal to

If $\sin C=\frac{\sin A}{2\cos B},$ then the triangle is

Let $y=\sin x·\sin ({60}^o+x)·\sin ({60}^o-x)$ then

$2\cos \frac{\mathrm{\pi }}{13}\cos \frac{9\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13}=$

$\cos 40°\cos 80°\cos 160°=$

If$\mathrm{ }\cos \left(\theta -\alpha \right),\cos \theta$ and $\cos \left(\theta +\alpha \right)$ are in HP, then $\cos \theta \sec \frac{\alpha }{2}$ is equal to

The expression ${\cos }^2\left(A-B\right)+{\cos }^{2}B-2\cos \left(A-B\right)\cos A\cos B$is

Value of ${\cos }^2(A-B)+{\cos }^{2}B-2\cos (A-B)\cos A\cos B$ is

$\frac{\cos x}{\cos (x-2y)}=\lambda \Rightarrow \tan (x-y)\tan y$ is equal to

The number of solutions of the equation $\sin x \cos 3x=\sin 3x\cos 5x$ in $\left[0, \frac{\pi }{2}\right]$ is

The value of $\sin \frac{\pi }{16}\sin \frac{3\pi }{16}\sin \frac{5\pi }{16}\sin \frac{7\pi }{16}$ is