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Question: Show that \(\tan 1{}^\circ \tan 2{}^\circ \tan 3{}^\circ .......................\tan 89{}^\circ =1\)...

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

Explanation

Solution

Hint: Try to simplify the left-hand side of the equation that we need to prove by using the property that tan⁡(90∘−α)=cot⁡α\tan \left( 90{}^\circ -\alpha \right)=\cot \alpha and cot⁡α=1tan⁡α\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∘tan⁡2∘tan⁡3∘.......................tan⁡89∘\tan 1{}^\circ \tan 2{}^\circ \tan 3{}^\circ .......................\tan 89{}^\circ
The expression can also be written as:
tan⁡1∘tan⁡2∘......tan44∘tan45∘tan⁡(90∘−44∘)......tan(90∘−2∘)tan⁡(90∘−1∘)\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⁡(90∘−α)=cot⁡α\tan \left( 90{}^\circ -\alpha \right)=\cot \alpha , we get
tan⁡1∘tan⁡2∘......tan44∘tan45∘cot⁡44∘......cot2∘cot⁡1∘\tan 1{}^\circ \tan 2{}^\circ ......tan44{}^\circ tan45{}^\circ \cot 44{}^\circ ......cot2{}^\circ \cot 1{}^\circ
Now we know that cot⁡α=1tan⁡α\cot \alpha =\dfrac{1}{\tan \alpha } . Using this in our expression, we get
tan⁡1∘tan⁡2∘......tan44∘tan45∘×1tan⁡44∘......1tan⁡2∘×1tan⁡1∘\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∘\tan 45{}^\circ .
∴1×tan⁡45∘=tan⁡45∘\therefore 1\times \tan 45{}^\circ =\tan 45{}^\circ
We know that the value of tan⁡45∘\tan 45{}^\circ is 1. Putting this in our expression, we get
tan⁡45∘=1\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∘tan⁡2∘tan⁡3∘.......................tan⁡89∘=1\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.