Question

The formula of \[\cot \left( {A - B} \right)\] is?

Answer 1

Hint: Here the question is related to trigonometry, we have to find the formula of cotangent or cot difference identity or formula. This can be find by using tangent difference identity \[\tan \left( {A - B} \right) = \dfrac{{\tan A - \tan B}}{{1 + \tan A \cdot \tan B}}\] , and further simplification using a reciprocal definition of trigonometric function we get the required solution.

Complete step by step solution:
Trigonometric ratios: Some ratios of the sides of a right-angle triangle with respect to its acute angle called trigonometric ratios of the angle.
The ratios defined are abbreviated as sin A, cos A, tan A, csc A or cosec A, sec A and cot A
These functions are defined as the reciprocal of the standard trigonometric functions: sine, cosine, and tangent, and hence they are called the reciprocal trigonometric functions.
The reciprocal trigonometric functions are: \[\dfrac{1}{{\sin A}} = \cos ec\,A\] , \[\dfrac{1}{{\cos A}} = sec\,A\] and \[\dfrac{1}{{\tan A}} = \cot \,A\] .
Consider the given question:
We need to find the formula of
 \[ \Rightarrow \cot \left( {A - B} \right)\]
Now, using the tangent or tan sum identity i.e., \[\tan \left( {A - B} \right) = \dfrac{{\tan A - \tan B}}{{1 + \tan A \cdot \tan B}}\] .
Take reciprocal of tan sum identity, then
 \[ \Rightarrow \dfrac{1}{{\tan \left( {A - B} \right)}} = \dfrac{{1 + \tan A \cdot \tan B}}{{\tan A - \tan B}}\]
By using a Reciprocal trigonometric function \[\dfrac{1}{{\tan A}} = \cot \,A\] or \[\dfrac{1}{{\cot A}} = \tan A\] , then we have
 \[ \Rightarrow \cot \left( {A - B} \right) = \dfrac{{1 + \dfrac{1}{{\cot A}} \cdot \dfrac{1}{{\cot B}}}}{{\dfrac{1}{{\cot A}} - \dfrac{1}{{\cot B}}}}\]
Take \[\cot A\cot B\] as LCM in both numerator and denominator of RHS, then
 \[ \Rightarrow \cot \left( {A - B} \right) = \dfrac{{\dfrac{{\cot A\cot B + 1}}{{\cot A\cot B}}}}{{\dfrac{{\cot B - \cot A}}{{\cot A\cot B}}}}\]
On cancelling the like terms in both numerator and denominator of RHS, then we get
 \[ \Rightarrow \cot \left( {A - B} \right) = \dfrac{{\cot A\cot B + 1}}{{\cot B - \cot A}}\]
Hence, the required formula of \[\cot \left( {A - B} \right) = \dfrac{{\cot A\cot B + 1}}{{\cot B - \cot A}}\] .
So, the correct answer is “ \[\cot \left( {A - B} \right) = \dfrac{{\cot A\cot B + 1}}{{\cot B - \cot A}}\] .”.

Note: The trigonometry ratios are sine, cosine, tangent, cosecant, secant and cotangent. These are abbreviated as sin, cos, tan, csc, sec and cot. We must know about the trigonometry definitions, identities and formulas which are involving the trigonometry ratios, while solving the trigonometry based questions.