Question

Find the value of ${{\left( \dfrac{1+i\tan \theta }{1-i\tan \theta } \right)}^{n}}-\dfrac{1+i\tan n\theta }{1-i\tan n\theta }$
$\left( A \right)\text{ 0}$
$\left( B \right)\text{ 1}$

Answer 1

Hint: In this question we have been given a trigonometric expression for which we have to find the value. We will solve this question by expanding the given expression by writing $\tan \theta $ in the form of $\dfrac{\sin \theta }{\cos \theta }$ and then taking the lowest common multiple. We will then use De moivre’s theorem which is given as ${{\left( \cos \theta +i\sin \theta \right)}^{n}}=\cos n\theta +i\sin n\theta $ and simplify the expression to get the required solution.

Complete step by step answer:
We have the expression given to us as:
 $\Rightarrow {{\left( \dfrac{1+i\tan \theta }{1-i\tan \theta } \right)}^{n}}-\dfrac{1+i\tan n\theta }{1-i\tan n\theta }$
Now we know that $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$ therefore, on substituting it in the expression, we get:
$\Rightarrow {{\left( \dfrac{1+i\dfrac{\sin \theta }{\cos \theta }}{1-i\dfrac{\sin \theta }{\cos \theta }} \right)}^{n}}-\dfrac{1+i\dfrac{\sin n\theta }{\cos n\theta }}{1-i\dfrac{\sin n\theta }{\cos n\theta }}$
On taking the lowest common multiple in the fractions, we get:
$\Rightarrow {{\left( \dfrac{\dfrac{\cos \theta +i\sin \theta }{\cos \theta }}{\dfrac{\cos \theta -i\sin \theta }{\cos \theta }} \right)}^{n}}-\dfrac{\dfrac{\cos n\theta +i\sin n\theta }{\cos n\theta }}{\dfrac{\cos n\theta -i\sin n\theta }{\cos n\theta }}$
On cancelling the terms, we get:
$\Rightarrow {{\left( \dfrac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta } \right)}^{n}}-\dfrac{\cos n\theta +i\sin n\theta }{\cos n\theta -i\sin n\theta }$
Now we know from De moivre’s theorem that ${{\left( \cos \theta +i\sin \theta \right)}^{n}}=\cos n\theta +i\sin n\theta $ therefore, on using in the above expression, we get:
$\Rightarrow \dfrac{\cos n\theta +i\sin n\theta }{\cos n\theta -i\sin n\theta }-\dfrac{\cos n\theta +i\sin n\theta }{\cos n\theta -i\sin n\theta }$
Now we can see that both the terms are the same therefore on simplifying, we get:
$\Rightarrow 0$, which is the required solution.

So, the correct answer is “Option A”.

Note: It is to be remembered that to add two or more fractions, the denominator of both them should be the same, if the denominator is not the same, the lowest common multiple known as L.C.M should be taken.
The various trigonometric identities and formulae should be remembered while doing these types of sums. the various Pythagorean identities should also be remembered while doing these type of questions.
To simplify any given equation, it is good practice to convert all the identities into $\sin $ and $\cos $ for simplifying.
If there is nothing to simplify, then only you should use the double angle formulas to expand the given equation.