Question

need help...
if A + B + C = π/2 radians show that
tan A tan B + tan B tan C + tan C tan A = 1

thanks!

Answer 1

\[\text{Because }C=\frac\pi2-(A+B)\]
Then :
\[\tan C=\frac1{\tan(A+B)}\]
So :
\[\tan A\tan B+\tan A\tan C+\tan B\tan C=
\\~~~~\tan A\tan B+\tan C(\tan A+\tan B)=
\\~~~~\tan A\tan B+(\tan A+\tan B)\times\frac1{\tan(A+B)}
\\~~~~\tan A\tan B+(\tan A+\tan B)\times\frac{1-\tan A\tan B}{\tan A+\tan B}
\\~~~~\tan A\tan B+1-\tan A\tan B=1\]

Answer 2

Smart! thank you so much!!

Answer 3

You are welcome !