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Question: If \[\cos {\text{ }}(\alpha {\text{ + }}\beta {\text{) = 0}}\], then \[{\text{sin }}(\alpha {\text{ ...

If cos⁡ (α + β) = 0\cos {\text{ }}(\alpha {\text{ + }}\beta {\text{) = 0}}, then sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}} can be reduced to
A. cos⁡β\cos \beta
B. cos⁡2β\cos 2\beta
C. sin⁡α\sin \alpha
D. sin⁡2α\sin 2\alpha

Explanation

Solution

Hint: To solve this question, we will use the trigonometric value of cos⁡900\cos {90^0} which is equal to zero. We will apply this in the given condition and from it we will find the value of α\alpha to find the value of sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}}.

Complete step-by-step answer:

Now, we are given cos⁡ (α + β) = 0\cos {\text{ }}(\alpha {\text{ + }}\beta {\text{) = 0}}. Now, at 900{90^0}, cos x = 0. So, we can write as

cos⁡ (α + β) = cos 900\cos {\text{ }}(\alpha {\text{ + }}\beta {\text{) = cos 9}}{{\text{0}}^0}

As, in the above equation, terms on both sides are of the form cos x. So, removing cos from the above equation, we get

α + β = 900\alpha {\text{ + }}\beta {\text{ = 9}}{{\text{0}}^0}. So, from this equation, we can find the value of α\alpha . So,

α = 900 - β\alpha {\text{ = 9}}{{\text{0}}^0}{\text{ - }}\beta

Now, we will put this value of α\alpha in sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}}, we get

sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}} = sin (900 - β - β){\text{sin }}({90^0}{\text{ - }}\beta {\text{ - }}\beta {\text{)}}

sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}} = sin⁡ (900 - 2β)\sin {\text{ (9}}{{\text{0}}^0}{\text{ - 2}}\beta {\text{)}}

Now, we know that for x lying in the first quadrant, we have sin⁡ (900 - x) = cos x \sin {\text{ (9}}{{\text{0}}^0}{\text{ - x) = cos x }}.

So, we can write as,

sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}} = sin⁡ (900 - 2β)\sin {\text{ (9}}{{\text{0}}^0}{\text{ - 2}}\beta {\text{)}} = cos⁡2β\cos 2\beta

Therefore, sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}} = cos⁡2β\cos 2\beta

So, option (B) is correct.

Note: When we come up with such types of questions, where we are given a trigonometric term on one side and a numerical value on the other side, we will always use the trigonometric values to find the solution. In this question, cos x can have a value equal to zero at 1800{180^0} so we get α + β = 1800\alpha {\text{ + }}\beta {\text{ = 18}}{{\text{0}}^0} and α = 1800 - β\alpha {\text{ = 18}}{{\text{0}}^0}{\text{ - }}\beta .When we put this value in sin (α - β){\text{sin }}(\alpha {\text{ - }}\beta {\text{)}}, we will get the same answer because in the second quadrant, sin⁡ (1800 - x) = cos x \sin {\text{ (18}}{{\text{0}}^0}{\text{ - x) = cos x }}.