Question

If $A+B+C=\frac{\pi }{2}$ then

• A. $\tan A\tan B+\tan B\tan C+\tan C\tan A=1$
• B. $\cot A+\cot B+\cot C=\cot A\cot B\cot C$
• C. $\cos 2A+\cos 2B+\cos 2C=1+4\sin A\sin B\sin C$
• D. All three are correct

Answer 1

Answer: (D)

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