Question

How do you find the exact value of \[\arctan \dfrac{\left( \arctan \left( \dfrac{9}{7} \right)-\arctan \left( \dfrac{7}{6} \right) \right)}{\left( \arctan \left( \dfrac{5}{3} \right)-\arctan \left( \dfrac{3}{2} \right) \right)}\]?

Answer 1

Hint: In the given question, we have been asked to find the exact value of the given inverse trigonometric expression. In order to solve the question, first we will need to simplify the given inverse trigonometric expression. Later by using a trigonometric special arc table or log table we will find the value of each individual term and then by putting these values, we will simplify the expression further. In this way we will get the exact value of the given expression.

Complete step by step solution:
We have given that,
\[\arctan \dfrac{\left( \arctan \left( \dfrac{9}{7} \right)-\arctan \left( \dfrac{7}{6} \right) \right)}{\left( \arctan \left( \dfrac{5}{3} \right)-\arctan \left( \dfrac{3}{2} \right) \right)}\]
Let,
\[A=\arctan \dfrac{\left( \arctan \left( \dfrac{9}{7} \right)-\arctan \left( \dfrac{7}{6} \right) \right)}{\left( \arctan \left( \dfrac{5}{3} \right)-\arctan \left( \dfrac{3}{2} \right) \right)}\]
Simplifying the above expression, we obtained
\[\tan A=\dfrac{\left( \arctan \left( \dfrac{9}{7} \right)-\arctan \left( \dfrac{7}{6} \right) \right)}{\left( \arctan \left( \dfrac{5}{3} \right)-\arctan \left( \dfrac{3}{2} \right) \right)}\]
Now,
Let,
\[x=\arctan \dfrac{9}{7}\Rightarrow \tan x=\dfrac{9}{7}\]
Using the trigonometric special arc table;
\[x={{\left( 52.15 \right)}^{0}}\]
Now,
Let,
\[y=\arctan \dfrac{7}{6}\Rightarrow \tan y=\dfrac{7}{6}\]
Using the trigonometric special arc table;
\[y={{\left( 49.39 \right)}^{0}}\]
Therefore,
\[\Rightarrow \arctan \dfrac{9}{7}-\arctan \dfrac{7}{6}=x-y={{\left( 52.15 \right)}^{0}}-{{\left( 49.39 \right)}^{0}}=\left( 2.73 \right)\]
Now,
Let,
\[u=\arctan \dfrac{5}{3}\Rightarrow \tan u=\dfrac{5}{3}\]
Using the trigonometric special arc table;
\[u={{\left( 59.03 \right)}^{0}}\]
Now,
Let,
\[v=\arctan \dfrac{3}{2}\Rightarrow \tan v=\dfrac{3}{2}\]
Using the trigonometric special arc table;
\[v={{\left( 56.30 \right)}^{0}}\]
Therefore,
\[\Rightarrow \arctan \dfrac{5}{3}-\arctan \dfrac{3}{2}=u-v={{\left( 59.03 \right)}^{0}}-{{\left( 56.30 \right)}^{0}}=\left( 2.73 \right)\]
Now,
We have
\[\tan A=\dfrac{\left( \arctan \left( \dfrac{9}{7} \right)-\arctan \left( \dfrac{7}{6} \right) \right)}{\left( \arctan \left( \dfrac{5}{3} \right)-\arctan \left( \dfrac{3}{2} \right) \right)}\]
Putting the values from above, we obtained
\[\tan A=\dfrac{2.73}{2.73}=1\]
Using the trigonometric ratios table, we know that
\[\tan \dfrac{\pi }{4}=1\]

Therefore, the exact value of \[\arctan \dfrac{\left( \arctan \left( \dfrac{9}{7} \right)-\arctan \left( \dfrac{7}{6} \right) \right)}{\left( \arctan \left( \dfrac{5}{3} \right)-\arctan \left( \dfrac{3}{2} \right) \right)}\]is \[\dfrac{\pi }{4}\].
Hence, it is the required answer.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.