Question

Find the value of log10(tan 1) + log10(tan 2) + + log10(tan 89).

Answer 1

Answer:

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Step-by-step explanation:

Given,

$log_{10}( \tan(1) ) + log_{10}( \tan(2) ) + ......... + log_{10}( \tan(89) )$

Now,

Let's denote,

$log_{10}(x) = log(x)$

Therefore,

We get,

$= log( \tan(1) ) + log( \tan(2) ) + ........ + log( \tan(89) )$

But,

We know that,

• $log(x) + log(y) = log(xy)$

Therefore,

We get,

$= log( \tan(1) \tan(2) ........... log( \tan(89) ) )$

But,

We know that,

• $\tan( \alpha ) = \cot(90 - \alpha )$

Therefore,

We get,

$= log( \tan(1) \cot(1) \tan(2) \cot(2) ................ \tan(45) )$

But,

We know that,

• $\tan( \alpha ) \times \cot( \alpha ) = 1$

Therefore,

We get,

$= log(1 \times 1 \times 1.......... \tan(45) ) \\ \\ = log( \tan(45) ) \\ \\ = log(1) \\ \\ = 0$