Prove that: $(1+\tan20)(1+\tan35)(1+\tan25)(1+\tan10)=4$ if $A+B+C=\pi$ and $\cos A = \cos B \, \cos C$.

Question

I have solved half of the problem by taking $\tan(A+B) = \tan(\pi -C)$. But I am stuck in the middle. So how to prove the statement ?

Answer 1

It is certainly true that

$1+\tan 10°=\dfrac{\sin 10° + \cos 10°}{\cos 10°}=(\sqrt{2})(\dfrac{\sin (10°+45°)}{\cos 10°})$

where $\sin (10°+45°)=(\sin 10°/\sqrt{2})+(\cos 10°/\sqrt{2})$ from the formula for the sine of a sum. Then, continuing:

$1+\tan 10°=(\sqrt{2})(\dfrac{\sin (10°+45°)}{\cos 10°})=(\sqrt{2})(\dfrac{\cos 35°}{\cos 10°})$

using $\sin (10°+45°)=\cos (90°-10°-45°)=\cos 35°$. Do the same with arguments of $20°, 25°, 35°$ in place of $10°$ and multiply the four resulting fractions together; all the trig functions cancel out of the product and you have just $(\sqrt{2})^4=4$.

The business with $A, B, C$, however, has me completely stumped. It does not enter the above equality at all!