Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$\mathrm{B}+\mathrm{C}=\frac{\pi }{2}-\mathrm{A}\Rightarrow \tan (\mathrm{B}+\mathrm{C})=\cot \mathrm{A}$ $\Rightarrow \frac{\tan \mathrm{B}+\tan \mathrm{C}}{1-\tan \mathrm{B}\tan \mathrm{C}}=\cot \mathrm{A}$ $\Rightarrow \tan \mathrm{A}\tan \mathrm{B}+\tan \mathrm{B}\tan \mathrm{C}+\tan \mathrm{C}\tan \mathrm{A}=1$ $\Rightarrow \tan \mathrm{A}\tan \mathrm{B}\tan \mathrm{C}(\cot \mathrm{C}+\cot \mathrm{A}+\cot \mathrm{B})=1$ $\Rightarrow \cot \mathrm{A}+\cot \mathrm{B}+\cot \mathrm{C}=\cot \mathrm{A}\cot \mathrm{B}\cot \mathrm{C}$ Again $\cos 2\mathrm{A}+\cos 2\mathrm{B}+\cos 2\mathrm{C}$ $=2\cos (\mathrm{A}+\mathrm{B})\cos (\mathrm{A}-\mathrm{B})+\cos 2\mathrm{C}$ $=2\cos \left(\frac{\pi }{2}-C\right)\cos (A-B)+1-2{\sin }^{2}C$ $=2\sin C[\cos (A-B)-\sin C]+1$ $=2\sin \mathrm{C}\left[\cos (\mathrm{A}-\mathrm{B})-\sin \frac{\pi }{2}-(\mathrm{A}+\mathrm{B})\right]+1$ $\begin{array}{l}=2\sin C[\cos (A-B)-\cos (A+B)]+1\\ =4\sin A\sin B\sin C+1\end{array}$