Question

If \[\alpha + \beta + \gamma = 2\pi \], then
1)\[\tan (\dfrac{\alpha }{2}) + \tan (\dfrac{\beta }{2}) + \tan (\dfrac{\gamma }{2}) = \tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2})\]
2)\[\tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2}) + \tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2}) + \tan (\dfrac{\gamma }{2})\tan (\dfrac{\alpha }{2}) = 1\]
3)\[\tan (\dfrac{\alpha }{2}) + \tan (\dfrac{\beta }{2}) + \tan (\dfrac{\gamma }{2}) = - \tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2})\]
4) None of these

Answer 1

Hint: This question needs the fundamental concepts of trigonometry. One should be aware of the trigonometric function to solve this question. Some properties of tan which help solve this question are:
The first property which is involved here is:
\[ \Rightarrow \tan (a + b + c) = = \dfrac{{\tan (a) + \tan (b) + \tan (c) - \tan (a)\tan (b)\tan (c)}}{{1 - \tan (a)\tan (b) - \tan (b)\tan (c) - \tan (c)\tan (a)}}\], where a, b, c are the angles
And the second property which is involved here is
\[ \Rightarrow \tan (n\pi ) = 0\], where n is a constant integer,
Apply these properties to approach the question and thereafter, solve it easily using simple algebra.

Complete step-by-step answer:
Let’s begin the question with the given condition, i.e.,
\[ \Rightarrow \alpha + \beta + \gamma = 2\pi \]
Now, by dividing by two on both the sides of the equation we get,
\[ \Rightarrow \dfrac{{\alpha + \beta + \gamma }}{2} = \pi \]
Now, taking by tangent function on both the sides of the equation we get,
\[ \Rightarrow \tan (\dfrac{{\alpha + \beta + \gamma }}{2}) = \tan (\pi )\]
Now, using the property of tangent, we simplify as mentioned above we get,\[ \Rightarrow \dfrac{{\tan (\dfrac{\alpha }{2}) + \tan (\dfrac{\beta }{2}) + \tan (\dfrac{\gamma }{2}) - \tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2})}}{{1 - \tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2}) - \tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2}) - \tan (\dfrac{\gamma }{2})\tan (\dfrac{\alpha }{2})}} = 0\]
Now, cross multiplying the equation we get,
\[ \Rightarrow \tan (\dfrac{\alpha }{2}) + \tan (\dfrac{\beta }{2}) + \tan (\dfrac{\gamma }{2}) - \tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2}) = 0\]
Now, by grouping all the positive terms on the left side and negative on the other we get,
\[ \Rightarrow \tan (\dfrac{\alpha }{2}) + \tan (\dfrac{\beta }{2}) + \tan (\dfrac{\gamma }{2}) = \tan (\dfrac{\alpha }{2})\tan (\dfrac{\beta }{2})\tan (\dfrac{\gamma }{2})\]
Thus, option(1) is the correct answer.
So, the correct answer is “Option 1”.

Note: This question requires the basic concepts of the tangent function. One should be well versed with those concepts before solving this question. Do not get intimidated by the equations involved in this question, they will be eliminated very easily afterwards. Calculation mistakes are possible in these questions, so try to avoid them and be sure of the final answer. Even after solving the question correctly, you might end up selecting the wrong answer as the options are very similar. So, be very careful while marking the right answer.