1987 AHSME Problems/Problem 20

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Problem

Evaluate $\log_{10}(\tan 1^{\circ})+\log_{10}(\tan 2^{\circ})+\log_{10}(\tan 3^{\circ})+\cdots+\log_{10}(\tan 88^{\circ})+\log_{10}(\tan 89^{\circ}).$

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2}\log_{10}(\frac{\sqrt{3}}{2}) \qquad \textbf{(C)}\ \frac{1}{2}\log_{10}2\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ \text{none of these}$

Solution

Because $\tan x \tan (90^\circ - x) = \tan x \cot x = 1$, $\tan 45^\circ = 1$, and $\log a + \log b = \log {ab}$, the answer is $\log_{10} {\tan 1^\circ \tan 2^\circ \dots \tan 89^\circ} = \log_{10} 1 = 0.$ $\boxed{\textbf{(A)}}.$

Solution 2

We have $\log{(\frac{\sin 1^\circ}{\cos 1^\circ} \times \frac{\sin 2^\circ}{\cos 2^\circ} \times ... \times \frac{\sin 89^\circ}{\cos 89^\circ})}$. However, $\sin{(x)} = \cos{(90^\circ - x)}$. Thus each pair of $\sin, \cos$ (for example, $\sin{1^\circ}, \cos{89^\circ}$) multiplies to $1$. Hence we have $\log{1} = 0$.

See also

1987 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
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