Question

The value of $ {\log _3}\tan 1^\circ + {\log _3}\tan 2^\circ + ........... + {\log _3}\tan 89^\circ $ is
A. $ 0 $
B. $ 1 $
C. $ 2 $
D. $ 3 $

Answer 1

Hint: Here we will use the logarithmic properties to simplify the given expression. We will use the Product rule: $ {\log _a}xy = {\log _a}x + {\log _a}y $ also apply the tangent and cot identity and then will simplify for the resultant value.

Complete step-by-step answer:
Take the given expression –
 $ {\log _3}\tan 1^\circ + {\log _3}\tan 2^\circ + ........... + {\log _3}\tan 89^\circ $
Apply the product rule in the above equation –
 $ \Rightarrow {\log _3}(\tan 1^\circ \times \tan 2^\circ \times ........... \times \tan 89^\circ ) $ ..... (A)
Now, use the trigonometric properties relating to tangent and the cot function.
 $ \tan 1^\circ = \tan (90^\circ - 89^\circ ) = \cot 89^\circ $
The above equation implies that -
 $ \tan 1^\circ \times \tan 89^\circ = 1 $ (Since tangent and the cot are inverse function and the product of tangent and cot is always equal to one)
Similarly for other terms in the given expression
 $ \tan 2^\circ \times \tan 88^\circ = 1 $
The above expression for any given angle can be expressed as –
 $ \tan A \times \tan (90^\circ - A) = 1 $
Since $ \tan (90^\circ - A) = \cot A $
And tangent and cot inverse functions so, the product is always one.
Place the above value in equation (A)
 $ \Rightarrow {\log _3}(\tan 1^\circ \times \tan 2^\circ \times ........... \times \tan 89^\circ ) $
 $ \Rightarrow {\log _3}(1) $
Log of one is always zero.
So, the correct answer is “Option A”.

Note: In other words, the logarithm is the power to which the number must be raised in order to get some other. Always remember the standard properties of the logarithm.... Product rule, quotient rule and the power rule. The basic logarithm properties are most important and the solution solely depends on it, so remember and understand its application properly. Be good in multiples and know the concepts of square and square root and apply accordingly.
Also refer to the below properties and rules of the logarithm.
Product rule: $ {\log _a}xy = {\log _a}x + {\log _a}y $
Quotient rule: $ {\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y $
Power rule: $ {\log _a}{x^n} = n{\log _a}x $
 Base rule: $ {\log _a}a = 1 $
Change of base rule: $ {\log _a}M = \dfrac{{\log M}}{{\log N}} $