Question

Find the value of \[{\tan ^{ - 1}}a + {\tan ^{ - 1}}b\] where \[a > 0,b > 0,ab > 1\] is equal to
A) \[{\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right)\]
B) \[{\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right) - \pi \]
C) \[\pi + {\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right)\]
D) \[{\text{none of these}}\]

Answer 1

Hint:
Here, we have to use the trigonometry identities to find the formula with the given conditions. First we will use the formula to find the sum of the inverse of tangent. Then we have to use the trigonometric ratio to evaluate for the given. The tangent function, along with sine and cosine, is one of the three basic trigonometric functions. In any right triangle, the tangent of an angle is the length of the opposite side (O) divided by the length of the adjacent side (A). In a formula, it is written simply as 'tan'.

Formula Used:
We will use the formula:
1) Trigonometric formula: \[{\tan ^{ - 1}}a + {\tan ^{ - 1}}b = \dfrac{{a + b}}{{1 - ab}}\]
2) Trigonometric Ratio: \[{\tan ^{ - 1}}\left( { - x} \right) = \pi + {\tan ^{ - 1}}x\]

Complete step by step solution:
Let us consider \[x = \dfrac{{a + b}}{{1 - ab}}\]
However, \[ab > 1\], then \[x\] will be negative.
By using \[{\tan ^{ - 1}}\left( { - x} \right) = \pi + {\tan ^{ - 1}}x\], we have
\[ \Rightarrow {\tan ^{ - 1}}\left( { - \left( {\dfrac{{a + b}}{{1 - ab}}} \right)} \right) = \pi + {\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right)\]
Therefore, \[{\tan ^{ - 1}}a + {\tan ^{ - 1}}b = \pi + {\tan ^{ - 1}}\left( {\dfrac{{a + b}}{{1 - ab}}} \right)\]

Note:
Cosine function and Sec functions are even functions; the remaining other functions are odd functions. Hence, the tan function is odd. At some angles the tangent function is undefined, and the problem is fundamental to drawing the graph of tangent function. We should Remember that the trigonometric ratios may be positive as well as negative. The application of tangent in daily life can be seen in the Architecture around us. The broadness and tallness of a building are the examples of the tangent. The examples of the tangent in daily life can be a school building, statue of liberty, bridges, monuments, pyramids etc. The values of tan θ and cot θ can have any value.