Question

What is the value of expression?
${\cos ^2}10 - \cos 10\cos 50 + {\cos ^2}50$
$
(a) {\text{ }}\dfrac{3}{4} \\
(b) {\text{ }}\dfrac{1}{3} \\
(c) {\text{ }}\dfrac{3}{2} \\
(d) {\text{ 0}} \\
$

Answer 1

Hint: In this question convert the $\cos {50^0}$terms into $\cos (60 - 10)$terms so that the basic trigonometric identity of $\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$ can be used, further basic identities like ${\sin ^2}x + {\cos ^2}x = 1$ will help solving the equation.

Complete Step-by-Step solution:
Given trigonometric equation is
${\cos ^2}10 - \cos 10\cos 50 + {\cos ^2}50$
Now the above equation is written as
$ \Rightarrow {\cos ^2}10 - \cos 10\cos \left( {60 - 10} \right) + {\cos ^2}\left( {60 - 10} \right)$
Now as we know $\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$ so use this property in above equation we have,
$ \Rightarrow {\cos ^2}10 - \cos 10\left[ {\cos 60\cos 10 + \sin 60\sin 10} \right] + {\left[ {\cos 60\cos 10 + \sin 60\sin 10} \right]^2}$
Now as we know $\cos 60 = \dfrac{1}{2},\sin 60 = \dfrac{{\sqrt 3 }}{2}$ so substitute these values we have,
$ \Rightarrow {\cos ^2}10 - \cos 10\left[ {\dfrac{1}{2} \times \cos 10 + \dfrac{{\sqrt 3 }}{2} \times \sin 10} \right] + {\left[ {\dfrac{1}{2} \times \cos 10 + \dfrac{{\sqrt 3 }}{2} \times \sin 10} \right]^2}$
Now simplify the above equation we have,
$ \Rightarrow {\cos ^2}10 - \dfrac{1}{2} \times {\cos ^2}10 - \dfrac{{\sqrt 3 }}{2} \times \sin 10\cos 10 + \left[ {\dfrac{1}{4} \times {{\cos }^2}10 + \dfrac{3}{4} \times {{\sin }^2}10 + \dfrac{{\sqrt 3 }}{2}\sin 10\cos 10} \right]$
Now as we know ${\sin ^2}10 = 1 - {\cos ^2}10$ so substitute this value in above equation and $\dfrac{{\sqrt 3 }}{2} \times \sin 10\cos 10$ term is cancel out so we have,
$ \Rightarrow {\cos ^2}10 - \dfrac{{{{\cos }^2}10}}{2} + \left[ {\dfrac{{{{\cos }^2}10}}{4} + \dfrac{3}{4} \times \left( {1 - {{\cos }^2}10} \right)} \right]$
Now again simplify the above equation we have,
$ \Rightarrow {\cos ^2}10 - \dfrac{{{{\cos }^2}10}}{2} + \dfrac{{{{\cos }^2}10}}{4} + \dfrac{3}{4} - \dfrac{{3{{\cos }^2}10}}{4}$
$ \Rightarrow {\cos ^2}10 - \left( {\dfrac{1}{2} - \dfrac{1}{4} + \dfrac{3}{4}} \right){\cos ^2}10 + \dfrac{3}{4}$
$ \Rightarrow {\cos ^2}10 - \left( {\dfrac{1}{2} + \dfrac{1}{2}} \right){\cos ^2}10 + \dfrac{3}{4}$
$ \Rightarrow {\cos ^2}10 - {\cos ^2}10 + \dfrac{3}{4}$
$ \Rightarrow \dfrac{3}{4}$
So this is the required answer.
Hence option (A) is correct.

Note: It is always advisable to remember the trigonometric identities. Although mugging up so many identities isn’t this easy but these identities are very useful for such formula based questions, as these also help saving a lot of time. So, keep practicing.