Question

If \[\tan n\theta = \tan m\theta \] , then find the different value of \[\theta \] will be in which progression.
A. A.P
B. G.P
C. H.P
D. None of these

Answer 1

Hints First obtain the general solution for the given equation, then substitute 1,2,3,… for n in the obtained general solution to observe the progression. Subtract the second term from the first term and then the third term from the second term to observe whether the difference is the same or not. If the difference is the same then the progression is in A.P.

Formula used
The general solution of \[\tan \theta = \tan \beta \] is \[\theta = n\pi + \beta ,n = 1,2,3,....\] .

Complete step by step solution
The given equation is \[\tan n\theta = \tan m\theta \].
Therefore, the general solution is,
\[n\theta = N\pi + m\theta \]
\[n\theta - m\theta = N\pi \]
\[\theta = \dfrac{{N\pi }}{{n - m}}\]
Now, substitute N=1,2,3,… for evaluation.
The sequence will be \[\left\{ {\dfrac{\pi }{{n - m}},\dfrac{{2\pi }}{{n - m}},\dfrac{{3\pi }}{{n - m}},...} \right\}\] .
Therefore, the common difference of second term and first term is \[\dfrac{{2\pi }}{{n - m}} - \dfrac{\pi }{{n - m}} = \dfrac{\pi }{{n - m}}\]
And third term and second term is \[\dfrac{{3\pi }}{{n - m}} - \dfrac{{2\pi }}{{n - m}} = \dfrac{\pi }{{n - m}}\], hence the sequence is in A.P.

The correct option is A.

Note Sometimes students write the general solution as \[n\theta = m\theta \] , hence they get \[\theta = 0\] as the answer but this is not correct. Use the general formula for \[\tan \theta = \tan \beta \] as \[\theta = n\pi + \beta ,n = 1,2,3,....\] and calculate to obtain the required answer.