Question

If $A=60{}^\circ \ and\ B=30{}^\circ $, verify that: $\tan \left( A-B \right)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}$.

Answer 1

Hint: We will be using the concept of trigonometric functions to solve the problem. We will first find the value of LHS by substituting the value of A and B. Then we will find the RHS by substituting the value of A and B and prove both the values to be equal.

Complete step-by-step answer:


Now, we have been given that$A=60{}^\circ \ and\ B=30{}^\circ $.

Now, we will first take LHS of the equation and substitute the value of A and B in it. So, we have,

$\begin{align}

  & \tan \left( A-B \right)=\tan \left( 60{}^\circ -30{}^\circ \right) \\

 & =\tan \left( 30{}^\circ \right) \\

\end{align}$

Now, we know that the value of $\tan \left( 30{}^\circ \right)=\dfrac{1}{\sqrt{3}}$.

Therefore, we have,

$\tan \left( A-B \right)=\dfrac{1}{\sqrt{3}}.............\left( 1 \right)$

Now, we will take RHS of the equation and substitute the value of A and B in it. So, we have,

$\dfrac{\tan A-\tan B}{1+\tan A\tan B}=\dfrac{\tan 60{}^\circ -\tan 30{}^\circ }{1+\tan 60{}^\circ \tan 30{}^\circ }$

Now, we know that the value of,

$\begin{align}

  & \tan 60{}^\circ =\sqrt{3} \\

 & \tan 30{}^\circ =\dfrac{1}{\sqrt{3}} \\

\end{align}$

So, we have the value as,

$\begin{align}

  & \dfrac{\tan A-\tan B}{1+\tan A\tan B}=\dfrac{\sqrt{3}-\dfrac{1}{\sqrt{3}}}{1+\sqrt{3}\times \dfrac{1}{\sqrt{3}}} \\

 & =\dfrac{\dfrac{3-1}{\sqrt{3}}}{+1} \\

 & =\dfrac{2}{2\times \sqrt{3}} \\

 & \dfrac{\tan A-\tan B}{1+\tan A\tan B}=\dfrac{1}{\sqrt{3}}..............\left( 2 \right) \\

\end{align}$

Now, from (1) and (2) we have that, if $A=60{}^\circ \ and\ B=30{}^\circ $ then$\tan \left( A-B \right)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}$.


Note: To solve these type of question it is important to note that to solve these type of questions on must remember the value of trigonometric ratios like

$\begin{align}

  & \tan 30{}^\circ =\dfrac{1}{\sqrt{3}} \\

 & \tan 60{}^\circ =\sqrt{3} \\

\end{align}$

Also, it has to be noted that to verify the given equation, we have simply substituted the values of A and B and find its value.