Algebranometry

Geometry Level 3

If $\left(1+\sqrt { 1+t } \right)\tan(t)=1+\sqrt { 1-t }$ for real $t$ and $f=\log _{ t }{ \sin(4t) }$ , find $(f+1)({ f }^{ 2 }+1)({ f }^{ 3 }+1)$ .

The answer is 8.

1 solution

Mark Hennings
Nov 27, 2018

We have (after some simplification) $\begin{aligned} \tan t & = \; \frac{1 + \sqrt{1-t}}{1 + \sqrt{1+t}} \\ \tan2t & = \; \frac{2\tan t}{1 - \tan^2t} \; = \; \frac{1 + \sqrt{1-t^2}}{t} \\ \sin4t & = \; \frac{2\tan2t}{1 + \tan^22t} \; = \; t \end{aligned}$ so that $f=1$ , making the answer $\boxed{8}$ .

Algebranometry

The answer is 8.

1 solution

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