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Question: Write the value of \[{{\tan }^{-1}}\dfrac{a}{b}-{{\tan }^{-1}}\dfrac{a-b}{a+b}\] ....

Write the value of tan⁡−1ab−tan⁡−1a−ba+b{{\tan }^{-1}}\dfrac{a}{b}-{{\tan }^{-1}}\dfrac{a-b}{a+b} .

Explanation

Solution

The above question is related to inverse trigonometric function and looking at the form of the equation, it is very clear that you have to use the formula tan⁡−1A−tan⁡−1B=tan⁡−1A−B1+AB{{\tan }^{-1}}A-{{\tan }^{-1}}B={{\tan }^{-1}}\dfrac{A-B}{1+AB} . After that you need to simplify the expression by taking the LCM and eliminating the possible terms. Finally you need to use the value tan⁡−11=45∘{{\tan }^{-1}}1=45{}^\circ to reach the final answer.

Complete step-by-step answer:
Before starting with the solution to the above question, we will first talk about the required details of different inverse trigonometric ratios. So, we must remember that inverse trigonometric ratios are completely different from trigonometric ratios and have many constraints related to their range and domain. So, to understand these constraints and the behaviour of inverse trigonometric functions, let us look at some of the important graphs. First, let us see the graph of sin⁡−1x{{\sin }^{-1}}x .

Now let us draw the graph of cos−1xco{{s}^{-1}}x .

Also, we will draw the graph of tan⁡−1x{{\tan }^{-1}}x as well.

So, looking at the above graphs, we can draw the conclusion that tan⁡−1x{{\tan }^{-1}}x is defined for all real values of x, i.e., the domain of the function tan⁡−1x{{\tan }^{-1}}x is all real numbers while its range comes out to be (−π2,π2)\left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right) . Unlike tan⁡−1x{{\tan }^{-1}}x the functions sin−1x and cos−1xsi{{n}^{-1}}x\text{ and co}{{\text{s}}^{-1}}x have the is defined only for x∈[−1,1]x\in [-1,1] .
Now moving to the solution to the above question, we will start with the simplification of the expression given in the question.
tan⁡−1ab−tan⁡−1a−ba+b{{\tan }^{-1}}\dfrac{a}{b}-{{\tan }^{-1}}\dfrac{a-b}{a+b}
Now, we know tan⁡−1A−tan⁡−1B=tan⁡−1A−B1+AB{{\tan }^{-1}}A-{{\tan }^{-1}}B={{\tan }^{-1}}\dfrac{A-B}{1+AB} , we get
tan⁡−1ab−a−ba+b1+a(a−b)b(a+b){{\tan }^{-1}}\dfrac{\dfrac{a}{b}-\dfrac{a-b}{a+b}}{1+\dfrac{a\left( a-b \right)}{b\left( a+b \right)}}
=tan⁡−1(a(a+b)−b(a−b)b(a+b)b(a+b)+a(a−b)b(a+b))={{\tan }^{-1}}\left( \dfrac{\dfrac{a\left( a+b \right)-b(a-b)}{b\left( a+b \right)}}{\dfrac{b(a+b)+a\left( a-b \right)}{b\left( a+b \right)}} \right)
=tan⁡−1(a2+ab−ab+b2a2+b2+ab−ab)={{\tan }^{-1}}\left( \dfrac{{{a}^{2}}+ab-ab+{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}+ab-ab} \right)
=tan⁡−1(a2+b2a2+b2)={{\tan }^{-1}}\left( \dfrac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)
=tan⁡−11={{\tan }^{-1}}1
We know that the value of tan⁡−11{{\tan }^{-1}}1 is equal to π4\dfrac{\pi }{4} . So, we can say that that the value of tan⁡−1ab−tan⁡−1a−ba+b{{\tan }^{-1}}\dfrac{a}{b}-{{\tan }^{-1}}\dfrac{a-b}{a+b} comes to be equal to π4\dfrac{\pi }{4} .

Note: While dealing with inverse trigonometric functions, it is preferred to know about the domains and ranges of the different inverse trigonometric functions. For example: the domain of sin⁡−1x{{\sin }^{-1}}x is [−1,1][-1,1] and the range is [−π2,π2]\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right] . Also be careful about the calculation and the signs while opening the brackets. The general mistake that a student can make is 1+x-(x-1)=1+x-x-1.