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Question: The value of \({{\tan }^{-1}}\left( \dfrac{a-b}{1+ab} \right)+{{\tan }^{-1}}\left( \dfrac{b-c}{1+bc}...

The value of tan⁡−1(a−b1+ab)+tan⁡−1(b−c1+bc){{\tan }^{-1}}\left( \dfrac{a-b}{1+ab} \right)+{{\tan }^{-1}}\left( \dfrac{b-c}{1+bc} \right), where a,b,c>0a,b,c>0
A) tan⁡−1a−tan⁡−1b{{\tan }^{-1}}a-{{\tan }^{-1}}b
B) tan⁡−1a−tan⁡−1c{{\tan }^{-1}}a-{{\tan }^{-1}}c
C) tan⁡−1b−tan⁡−1c{{\tan }^{-1}}b-{{\tan }^{-1}}c
D) tan⁡−1c−tan⁡−1a{{\tan }^{-1}}c-{{\tan }^{-1}}a

Explanation

Solution

First, assume the value of a, b and c in terms of tan. After that apply the formula tan⁡(A−B)=tan⁡A−tan⁡B1−tan⁡Atan⁡B\tan \left( A-B \right)=\dfrac{\tan A-\tan B}{1-\tan A\tan B}. Then cancel out tan and tan⁡−1{{\tan }^{-1}}. Now, substitute back the value of assumed value. The value obtained is the desired result.

Complete step by step answer:
Let a=tan⁡xa=\tan x, b=tan⁡yb=\tan y and c=tan⁡zc=\tan z
Substitute the values in the equation,
⇒tan⁡−1(tan⁡x−tan⁡y1+tan⁡xtan⁡y)+tan⁡−1(tan⁡y−tan⁡z1+tan⁡ytan⁡z)\Rightarrow {{\tan }^{-1}}\left( \dfrac{\tan x-\tan y}{1+\tan x\tan y} \right)+{{\tan }^{-1}}\left( \dfrac{\tan y-\tan z}{1+\tan y\tan z} \right)
As, we know that, the formula of tan (A-B),
⇒tan⁡(A−B)=tan⁡A−tan⁡B1−tan⁡Atan⁡B\Rightarrow \tan \left( A-B \right)=\dfrac{\tan A-\tan B}{1-\tan A\tan B}
Substitute the value according to the formula,
⇒tan⁡−1[tan⁡(x−y)]+tan⁡−1[tan⁡(y−z)]\Rightarrow {{\tan }^{-1}}\left[ \tan \left( x-y \right) \right]+{{\tan }^{-1}}\left[ \tan \left( y-z \right) \right]
Cancel out tan and tan-1 from the equation,
⇒x−y+y−z\Rightarrow x-y+y-z
Add or subtract the like terms,
⇒x−z\Rightarrow x-z
Now, substitute the value of x and z,
⇒tan⁡−1a−tan⁡−1c\Rightarrow {{\tan }^{-1}}a-{{\tan }^{-1}}c
Thus, the value of tan⁡−1(a−b1+ab)+tan⁡−1(b−c1+bc){{\tan }^{-1}}\left( \dfrac{a-b}{1+ab} \right)+{{\tan }^{-1}}\left( \dfrac{b-c}{1+bc} \right) is tan⁡−1a−tan⁡−1c{{\tan }^{-1}}a-{{\tan }^{-1}}c.

Hence, option (B) is the correct answer.

Note:
Trigonometry is an important branch of Mathematics. It mainly deals with triangles and their angles. It provides the relationships between the lengths and angles of triangles. It is the study of the relationships which involve angles, lengths, and heights of triangles.
Trigonometric formulas involve many trigonometric functions. These formulas and identities are true for all possible values of the variables. Trigonometric Ratios are also very basic to provide the relationship between the measurement of the angles and the length of the side of the right-angled triangle.
The six ratios which are the core of trigonometry are Sine (sin), Cosine (cos), Tangent (tan), Secant (sec), Cosecant (cosec), and Cotangent (cot).