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Question: Simplify \[\dfrac{\cos 37{}^\circ }{\sin 53{}^\circ }\times \dfrac{\sin 18{}^\circ }{\cos 72{}^\circ...

Simplify cos⁡37∘sin⁡53∘×sin⁡18∘cos⁡72∘\dfrac{\cos 37{}^\circ }{\sin 53{}^\circ }\times \dfrac{\sin 18{}^\circ }{\cos 72{}^\circ } using proper formula.

Explanation

Solution

Hint: Given equation has 2 terms with cosine and sine in both the terms. So, First solve each term separating from each other. Convert everything into cosine or sine in each term and then multiply them together, as it will be easy to cancel or substitute if everything will be in the same terms.
By basic knowledge of trigonometry, we can say that these are true
sin⁡(90−x)=cos⁡x\sin \left( 90-x \right)=\cos x, cos⁡(90−x)=sin⁡x\cos \left( 90-x \right)=\sin x

Complete step-by-step answer:
Given expression in the question which is to be solved is:
cos⁡37∘sin⁡53∘×sin⁡18∘cos⁡72∘\dfrac{\cos 37{}^\circ }{\sin 53{}^\circ }\times \dfrac{\sin 18{}^\circ }{\cos 72{}^\circ }
By dividing the equation into two terms, we write divided equation as
(cos⁡37∘sin⁡53∘)×(sin⁡18∘cos⁡72∘)\left( \dfrac{\cos 37{}^\circ }{\sin 53{}^\circ } \right)\times \left( \dfrac{\sin 18{}^\circ }{\cos 72{}^\circ } \right)
Let the first term be designated as A.
Let the second term of expression be represented as B.
By above we can say that the values of A,B:
A=cos⁡37∘sin⁡53∘A=\dfrac{\cos 37{}^\circ }{\sin 53{}^\circ } and B=sin⁡18∘cos⁡72∘B=\dfrac{\sin 18{}^\circ }{\cos 72{}^\circ }
Solving of A:
By above equations we can say value of A is:
A=cos⁡37∘sin⁡53∘A=\dfrac{\cos 37{}^\circ }{\sin 53{}^\circ }
By converting the degrees into radian we get A to be:
A=cos⁡(37π180)sin⁡(37π180)A=\dfrac{\cos \left( \dfrac{37\pi }{180} \right)}{\sin \left( \dfrac{37\pi }{180} \right)}
By observing carefully we can write A also as follows:
A=cos⁡(π2−53π180)sin⁡(53π180)A=\dfrac{\cos \left( \dfrac{\pi }{2}-\dfrac{53\pi }{180} \right)}{\sin \left( \dfrac{53\pi }{180} \right)}
By basic knowledge of trigonometry cos⁡(90−x)=sin⁡x\cos \left( 90-x \right)=\sin x
By applying this to the above equation of A, we get
B=sin⁡(53π180)sin⁡(53π180)=1B=\dfrac{\sin \left( \dfrac{53\pi }{180} \right)}{\sin \left( \dfrac{53\pi }{180} \right)}=1
Solving of B: By above equations we say value of B is
B=sin⁡18∘cos⁡72∘B=\dfrac{\sin 18{}^\circ }{\cos 72{}^\circ }
By converting into radian we get B to be as follow. By carefully observing we can also write B in the form of:
sin⁡(π2−72π180)cos⁡(72π180)\dfrac{\sin \left( \dfrac{\pi }{2}-\dfrac{72\pi }{180} \right)}{\cos \left( \dfrac{72\pi }{180} \right)}
By basic knowledge of trigonometry sin⁡(90−x)=cos⁡x\sin \left( 90-x \right)=\cos x
By applying this to the above equation of B, we get
B=cos⁡(72π180)cos⁡(72π180)=1B=\dfrac{\cos \left( \dfrac{72\pi }{180} \right)}{\cos \left( \dfrac{72\pi }{180} \right)}=1
Required equation is A⋅BA\cdot B
By substituting A, B values we get A⋅B=1A\cdot B=1
Therefore, 1 is the value of required expression.

Note: Alternate way: - Here, we converted numerator in terms of denominator. We can also convert denominators in terms of numerator and get the same result.