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Question: Simplify the expression \[\dfrac{{\sin a\cos b + \cos a\sin b}}{{\cos a\cos b - \sin a\sin b}} \time...

Simplify the expression sin⁡acos⁡b+cos⁡asin⁡bcos⁡acos⁡b−sin⁡asin⁡b×cos⁡acos⁡b+sin⁡asin⁡bsin⁡acos⁡b−cos⁡asin⁡b\dfrac{{\sin a\cos b + \cos a\sin b}}{{\cos a\cos b - \sin a\sin b}} \times \dfrac{{\cos a\cos b + \sin a\sin b}}{{\sin a\cos b - \cos a\sin b}}.

Explanation

Solution

Hint : We have to simplify the given trigonometric expression. We solve this question using the concept of the various formulas of the trigonometric function. We should have the knowledge of the relation between tangent, sine and cosine function. First, we will simplify the given trigonometric expression using the formula of the sine of sum and difference of two angles and the cosine of the sum and difference of two angles. Then we will simplify the expression using the relation of the tangent function, and hence we will get the required simplified expression.

Complete step-by-step answer :
Given:
The expression is sin⁡acos⁡b+cos⁡asin⁡bcos⁡acos⁡b−sin⁡asin⁡b×cos⁡acos⁡b+sin⁡asin⁡bsin⁡acos⁡b−cos⁡asin⁡b\dfrac{{\sin a\cos b + \cos a\sin b}}{{\cos a\cos b - \sin a\sin b}} \times \dfrac{{\cos a\cos b + \sin a\sin b}}{{\sin a\cos b - \cos a\sin b}}.
As, we know that the formula of sine of difference of two angles is given as:
sin⁡(a−b)=sin⁡acos⁡b−sin⁡bcos⁡a\sin \left( {a - b} \right) = \sin a\cos b - \sin b\cos a
We also know that the formula for cosine of difference of two angles is given as:
cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos \left( {a - b} \right) = \cos a\cos b + \sin a\sin b
As, we know that the formula of sine of sum of two angles is given as:
sin⁡(a+b)=sin⁡acos⁡b+sin⁡bcos⁡a\sin \left( {a + b} \right) = \sin a\cos b + \sin b\cos a
We also know that the formula for cosine of sum of two angles is given as:
cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b
Using the above formulas, we can write the expression as:
sin⁡acos⁡b+cos⁡asin⁡bcos⁡acos⁡b−sin⁡asin⁡b×cos⁡acos⁡b+sin⁡asin⁡bsin⁡acos⁡b−cos⁡asin⁡b=sin⁡(a+b)cos⁡(a+b)×cos⁡(a−b)sin⁡(a−b)\dfrac{{\sin a\cos b + \cos a\sin b}}{{\cos a\cos b - \sin a\sin b}} \times \dfrac{{\cos a\cos b + \sin a\sin b}}{{\sin a\cos b - \cos a\sin b}} = \dfrac{{\sin \left( {a + b} \right)}}{{\cos \left( {a + b} \right)}} \times \dfrac{{\cos \left( {a - b} \right)}}{{\sin \left( {a - b} \right)}}
Now, we also know that the relation between tangent, sine and cosine function are given as:
tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}}
Using the relation, we can write the expression as:
sin⁡acos⁡b+cos⁡asin⁡bcos⁡acos⁡b−sin⁡asin⁡b×cos⁡acos⁡b+sin⁡asin⁡bsin⁡acos⁡b−cos⁡asin⁡b=tan⁡(a+b)tan⁡(a−b)\dfrac{{\sin a\cos b + \cos a\sin b}}{{\cos a\cos b - \sin a\sin b}} \times \dfrac{{\cos a\cos b + \sin a\sin b}}{{\sin a\cos b - \cos a\sin b}} = \dfrac{{\tan \left( {a + b} \right)}}{{\tan \left( {a - b} \right)}}
Hence, the simplified expression of the function sin⁡acos⁡b+cos⁡asin⁡bcos⁡acos⁡b−sin⁡asin⁡b×cos⁡acos⁡b+sin⁡asin⁡bsin⁡acos⁡b−cos⁡asin⁡b\dfrac{{\sin a\cos b + \cos a\sin b}}{{\cos a\cos b - \sin a\sin b}} \times \dfrac{{\cos a\cos b + \sin a\sin b}}{{\sin a\cos b - \cos a\sin b}} is tan⁡(a+b)tan⁡(a−b)\dfrac{{\tan \left( {a + b} \right)}}{{\tan \left( {a - b} \right)}}.
So, the correct answer is tan⁡(a+b)tan⁡(a−b)\dfrac{{\tan \left( {a + b} \right)}}{{\tan \left( {a - b} \right)}}”.

Note : We can also simplify the given trigonometric expression using the formula of the tangent of the sum of two angles and the tangent of the difference of two angles. To solve these types of questions, we try to simplify the expression such that we get the relation of the expression in one of the formulas of the trigonometric functions. Therefore, we should keep in mind all the formulas while doing these questions.