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Question: Find the value of \[\dfrac{{\cos 75^\circ \cdot \sin 12^\circ \cdot \cos 18^\circ }}{{\sin 15^\circ ...

Find the value of cos⁡75∘⋅sin⁡12∘⋅cos⁡18∘sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘\dfrac{{\cos 75^\circ \cdot \sin 12^\circ \cdot \cos 18^\circ }}{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }}.
A.22
B.11
C.00
D.−1 - 1

Explanation

Solution

Here, we will use the basic identities of the trigonometric functions to find out the value of the given equation. We will first convert cosine function in the numerator in terms of sine function and sine function in terms of cosine function using the properties of the trigonometric function. We will further simplify the fraction to find the required value.

Complete step-by-step answer:
We have to simplify the given equation by using the properties of trigonometric functions.
Firstly we will simplify the given equation by writing the trigonometry functions in the numerator of the equation in the form of sin⁡(90∘−θ)\sin \left( {90^\circ - \theta } \right) or cos⁡(90∘−θ)\cos \left( {90^\circ - \theta } \right).
We can write cos⁡75∘=cos⁡(90∘−15∘)\cos 75^\circ = \cos \left( {90^\circ - 15^\circ } \right),sin⁡12∘=sin⁡(90∘−78∘)\sin 12^\circ = \sin \left( {90^\circ - 78^\circ } \right) and cos⁡18∘=cos⁡(90∘−72∘)\cos 18^\circ = \cos \left( {90^\circ - 72^\circ } \right). Therefore, we get
⇒cos⁡75∘⋅sin⁡12∘⋅cos⁡18∘sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘=cos⁡(90∘−15∘)⋅sin⁡(90∘−78∘)⋅cos⁡(90∘−72∘)sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘\Rightarrow \dfrac{{\cos 75^\circ \cdot \sin 12^\circ \cdot \cos 18^\circ }}{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }} = \dfrac{{\cos \left( {90^\circ - 15^\circ } \right) \cdot \sin \left( {90^\circ - 78^\circ } \right) \cdot \cos \left( {90^\circ - 72^\circ } \right)}}{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }}
Also, we know that cos⁡(90∘−θ)=sin⁡θ\cos \left( {90^\circ - \theta } \right) = \sin \theta and sin⁡(90∘−θ)=cos⁡θ\sin \left( {90^\circ - \theta } \right) = \cos \theta as the entire trigonometry functions is positive in the first quadrant. Therefore the equation becomes
⇒cos⁡75∘⋅sin⁡12∘⋅cos⁡18∘sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘=sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘\Rightarrow \dfrac{{\cos 75^\circ \cdot \sin 12^\circ \cdot \cos 18^\circ }}{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }} = \dfrac{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }}{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }}
Canceling out similar terms, we get
⇒cos⁡75∘⋅sin⁡12∘⋅cos⁡18∘sin⁡15∘⋅cos⁡78∘⋅sin⁡72∘=1\Rightarrow \dfrac{{\cos 75^\circ \cdot \sin 12^\circ \cdot \cos 18^\circ }}{{\sin 15^\circ \cdot \cos 78^\circ \cdot \sin 72^\circ }} = 1
Hence, the value of the equation is 11.
So, option B is the correct option.

Note: We should know the different properties of the trigonometric function in order to solve the question easily. It is also important for us to keep in mind the quadrant which all functions is positive or negative as in the first quadrant all the functions. In the second quadrant, only sine and cosecant are positive. In the third quadrant, only the tangent and cotangent function is positive. And in the fourth quadrant, only cosine and secant functions are positive.