Question

Show that $\tan 1{}^\circ \tan 2{}^\circ \tan 3{}^\circ .......................\tan 89{}^\circ =1$ .

Answer 1

Hint: Try to simplify the left-hand side of the equation that we need to prove by using the property that $\tan \left( 90{}^\circ -\alpha \right)=\cot \alpha $ and $\cot \alpha =\dfrac{1}{\tan \alpha }$ .

Complete step-by-step answer:
Now we will start with the simplification of the left-hand side of the equation that is given in the question which we are asked to prove.
$\tan 1{}^\circ \tan 2{}^\circ \tan 3{}^\circ .......................\tan 89{}^\circ $
The expression can also be written as:
$\tan 1{}^\circ \tan 2{}^\circ ......tan44{}^\circ tan45{}^\circ \tan \left( 90{}^\circ -44{}^\circ \right)......tan\left( 90{}^\circ -2{}^\circ \right)\tan \left( 90{}^\circ -1{}^\circ \right)$
Now using the property of tangent that $\tan \left( 90{}^\circ -\alpha \right)=\cot \alpha $ , we get
$\tan 1{}^\circ \tan 2{}^\circ ......tan44{}^\circ tan45{}^\circ \cot 44{}^\circ ......cot2{}^\circ \cot 1{}^\circ $
Now we know that $\cot \alpha =\dfrac{1}{\tan \alpha }$ . Using this in our expression, we get
$\tan 1{}^\circ \tan 2{}^\circ ......tan44{}^\circ tan45{}^\circ \times \dfrac{1}{\tan 44{}^\circ }......\dfrac{1}{\tan 2{}^\circ }\times \dfrac{1}{\tan 1{}^\circ }$
So, we can see in the above expression that all the terms are getting cancelled except one term, i.e., $\tan 45{}^\circ $ .
$\therefore 1\times \tan 45{}^\circ =\tan 45{}^\circ $
We know that the value of $\tan 45{}^\circ $ is 1. Putting this in our expression, we get
$\tan 45{}^\circ =1$
As we have shown that the left-hand side of the equation given in the question is equal to the right-hand side of the equation in the question, which is equal to 1. Hence, we can say that we have proved that $\tan 1{}^\circ \tan 2{}^\circ \tan 3{}^\circ .......................\tan 89{}^\circ =1$.


Note: Be careful about the calculation and the signs while opening the brackets. The general mistake that a student can make is 1+x-(x-1)=1+x-x-1. Also, you need to remember the properties related to complementary angles and trigonometric ratios. It is preferred that while dealing with questions as above, you must first try to observe the pattern of the consecutive terms before applying the formulas, as directly applying the formulas may complicate the question.