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Question: Solve \[{\cos ^2}\left( {\dfrac{\pi }{6} + \theta } \right) - {\sin ^2}\left( {\dfrac{\pi }{6} - \th...

Solve cos⁡2(π6+θ)−sin⁡2(π6−θ)={\cos ^2}\left( {\dfrac{\pi }{6} + \theta } \right) - {\sin ^2}\left( {\dfrac{\pi }{6} - \theta } \right) =
A.12cos⁡2θ\dfrac{1}{2}\cos 2\theta
B.00
C.−12cos⁡2θ- \dfrac{1}{2}\cos 2\theta
D.12\dfrac{1}{2}

Explanation

Solution

Hint : This question is related directly to the identity formula of trigonometric ratios . We will use the formula of cos⁡2A−sin⁡2B=cos⁡(A+B)cos⁡(A+B){\cos ^2}A - {\sin ^2}B = \cos \left( {A + B} \right)\cos \left( {A + B} \right) . These are the identity of compound angles of trigonometric ratios . . A compound angle is an algebraic sum of two or more angles. We use trigonometric identities to connote compound angles through trigonometric functions. The sum and difference of functions in trigonometry can be solved using the compound angle formula or the addition formula . You have to remember the different formulas so as to apply them in questions accordingly .

Complete step-by-step answer :
Given : cos⁡2(π6+θ)−sin⁡2(π6−θ){\cos ^2}\left( {\dfrac{\pi }{6} + \theta } \right) - {\sin ^2}\left( {\dfrac{\pi }{6} - \theta } \right)
Now , using the identity cos⁡2A−sin⁡2B=cos⁡(A+B)cos⁡(A+B){\cos ^2}A - {\sin ^2}B = \cos \left( {A + B} \right)\cos \left( {A + B} \right) , we get
=cos⁡(π6+θ+π6−θ)cos⁡(π6+θ−π6+θ)= \cos \left( {\dfrac{\pi }{6} + \theta + \dfrac{\pi }{6} - \theta } \right)\cos \left( {\dfrac{\pi }{6} + \theta - \dfrac{\pi }{6} + \theta } \right)
On simplifying we get .
=cos⁡(2π6)cos⁡(2θ)= \cos \left( {\dfrac{{2\pi }}{6}} \right)\cos \left( {2\theta } \right)
On solving we get .
=cos⁡(π3)cos⁡(2θ)= \cos \left( {\dfrac{\pi }{3}} \right)\cos \left( {2\theta } \right) , now putting the value of cos⁡(π3)=12\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2} .
=12cos⁡(2θ)= \dfrac{1}{2}\cos \left( {2\theta } \right) .
Therefore , option (A ) is the correct answer for the given question .
So, the correct answer is “Option A”.

Note : Trigonometry is all about angles and their measurement . When discussing the various trigonometric functions, we keep in mind the formula of compound angles to give accurate results. A compound cut comprises two angles . They are the bevel angle and the miter angle. The bevel angle (or blade tilt) is basically the tilt of the saw blade from vertical on the saw table. Similarly , the miter angle is set on the meter gauge of the table saw. Moreover, a perpendicular cut has a meter of 0∘{0^ \circ } . You also have to remember the values of trigonometric ratios at different angles . These questions are short and tricky .