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Question: The value of trigonometric function \[\sin {{37}^{\circ }}\sin {{23}^{\circ }}-\cos {{37}^{\circ }}\...

The value of trigonometric function sin⁡37∘sin⁡23∘−cos⁡37∘cos⁡23∘\sin {{37}^{\circ }}\sin {{23}^{\circ }}-\cos {{37}^{\circ }}\cos {{23}^{\circ }} is equal to:
A. cos⁡14∘\cos {{14}^{\circ }}
B. cos⁡60∘\cos {{60}^{\circ }}
C. −cos⁡14∘-\cos {{14}^{\circ }}
D. −cos⁡60∘-\cos {{60}^{\circ }}

Explanation

Solution

Hint:The terms in the above question are very similar and by looking closely and carefully, we can see that they can be solved using the trigonometric identity which is as follows: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B

Complete step-by-step answer:
In the question, we have been given the expression as,
sin⁡37∘sin⁡23∘−cos⁡37∘cos⁡23∘\sin {{37}^{\circ }}\sin {{23}^{\circ }}-\cos {{37}^{\circ }}\cos {{23}^{\circ }}
On taking -1 common from the expression, we get as follows:
−(cos⁡37∘cos⁡23∘−sin⁡37∘sin⁡23∘)-\left( \cos {{37}^{\circ }}\cos {{23}^{\circ }}-\sin {{37}^{\circ }}\sin {{23}^{\circ }} \right)
Now, we know that the trigonometric identity of cos⁡(A+B)\cos \left( A+B \right) is given as follows:
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B
Let us compare the expression given to us in the question and the above identity. Then, we can see that in the above expression, we have been given A=37∘A={{37}^{\circ }} and B=23∘B={{23}^{\circ }}.
So by using the trigonometric identity in the given expression, we get as follows:
−(cos⁡37∘cos⁡23∘−sin⁡37∘sin⁡23∘)=−[cos⁡(37∘+23∘)]-\left( \cos {{37}^{\circ }}\cos {{23}^{\circ }}-\sin {{37}^{\circ }}\sin {{23}^{\circ }} \right)=-\left[ \cos \left( {{37}^{\circ }}+{{23}^{\circ }} \right) \right]
Adding the angles, we get as follows:
−[cos⁡(37∘+23∘)]=−cos⁡60∘-\left[ \cos \left( {{37}^{\circ }}+{{23}^{\circ }} \right) \right]=-\cos {{60}^{\circ }}
Therefore the correct answer of the given question is option D.

Note: Don’t forget to take -1 as common. Then use the trigonometric identity of cos⁡(A+B)\cos \left( A+B \right) as we know that cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B. If we don’t consider the signs properly, we might choose option B in a hurry. Also don’t make the mistake of the algebraic signs while using the identity of cos⁡(A+B)\cos \left( A+B \right). If we use the incorrect trigonometric identity by mistake, as cos⁡(A−B)=sin⁡Asin⁡B−cos⁡Acos⁡B\cos \left( A-B \right)=\sin A\sin B-\cos A\cos B so we might end up getting the answer as cos⁡14∘\cos {{14}^{\circ }}. So we might end up choosing the option as A or C.Students should remember important trigonometric formulas,identities and standard angles to solve these types of questions.