Trigonometric Functions of Sum and Difference of Two Angles

Let  $\mathrm{f}\left(\mathrm{x}\right)={\sin }^6\mathrm{x}+{\cos }^6\mathrm{x}+\mathrm{k}({\sin }^4\mathrm{x}+{\cos }^4\mathrm{x})$  for some real number $\mathrm{k},$ then the value of $\mathrm{k}$ for which $\mathrm{f}\left(\mathrm{x}\right)$ is constant for all values of $\mathrm{x}$ is

$\left(1+\cos \frac{\pi }{8}\right)\left(1+\cos \frac{\mathrm{}3\pi }{8}\right)\left(1+\cos \frac{\mathrm{}5\pi }{8}\right)\left(1+\cos \frac{\mathrm{}7\pi }{8}\right)$ is equal to –

If $\alpha +\beta +\gamma =2\pi$, then

If  $\alpha +\beta +\gamma =\mathrm{}\mathrm{}\mathrm{}2\pi ,\mathrm{then}$ –

If $\theta$ is acute and $\sin \theta =\cos 2\theta$, then $(\sin \theta +\cos \theta )=?$

For a $∆PQR$, if $\cos P+\cos Q+\cos R=2-\cos Q$, then value of $\frac{p+r}{q}$ equals, (where $p, q, r$ are side lengths of $∆PQR$)

The value of the expression $3\left[{\sin }^4\left(\frac{3\mathrm{\pi }}{2}-\alpha \right)+{\sin }^4(3\pi +\alpha )\right]-2\left[{\sin }^6\left(\frac{\mathrm{\pi }}{2}+\alpha \right)+{\sin }^6(5\pi -\alpha )\right]$ is

In a triangle $ABC$ if $cot\frac{A}{2}, \tan \frac{B}{2}, cot\frac{C}{2}$ are in harmonic progression, then the value of $1-\tan \frac{A}{2}\tan \frac{C}{2}$ is, where $\tan \frac{A}{2}+\tan \frac{C}{2}\ne 0$

If $\cot 1^{°}\cot 2^{°}\cot 3^{°}...\cot {89}^{°}=4\mathrm{x}-{\mathrm{x}}^2$, then value of $\mathrm{x}$ equals to

If $y=\frac{45-10\cos 2\theta -30\cos 4\theta -5\cos 6\theta }{\sin 5\theta +7\sin 3\theta +9\sin \theta }$, then $y$ is

If $0<A,B<\frac{\mathrm{\pi }}{4}, \sin \left(A-B\right)=\frac{1}{\sqrt{10}}$ and $\cos \left(A+B\right)=\frac{2}{\sqrt{29}}$, then the value of $\left[{\log }_2\left(\tan 2A\right)\right]$ (where $\left[x\right]$ denotes the greatest integer function of $x$) is

If $\sum _{r=2021}^{2035}\frac{1}{\sin \left(r+\lambda \right) \sin \left(r+\lambda +1\right)}=\frac{\cot \left(\lambda +a\right)-\cot \left(\lambda +b\right)}{\sin c}$, then value of $\frac{b-a}{5c}$ is

$4\left({\sin }^{6}x+{\cos }^{6}x\right)+6{\left(\sin x+\cos x\right)}^2+3{\left(\sin x-\cos x\right)}^4-10$ is equal to

If $(1+\tan A)(1+\tan B)=2$, then $\sin \left(A+B\right)=?$

If $\tan \mathrm{\alpha }$ and $\tan \mathrm{\beta }$ are the roots of the equation $x^2+ bx+\left(c+1\right)=0,$ then the value of $\cos 2\left(\mathrm{\alpha }+\mathrm{\beta }\right)$ is