Question

What is the value of \[\cos \left( {2{{\cos }^{ - 1}}0.8} \right)\]?
A.\[0.81\]
B.\[0.56\]
C.\[0.48\]
D.\[0.28\]

Answer 1

Hint First, we will first assume \[2{\cos ^{ - 1}}0.8 = \theta \] and then use this value in the property of trigonometric functions, \[1 + \cos \theta = 2{\cos ^2}\left( {\dfrac{\theta }{2}} \right)\]. Then we will substitute the obtained value of \[\theta \] in the given equation to find the required answer.

Complete step-by-step answer:
We are given \[\cos \left( {2{{\cos }^{ - 1}}0.8} \right)\].
Let us assume that \[2{\cos ^{ - 1}}0.8 = \theta \].
Dividing the assumed equation by 2 on both sides, we get
\[
   \Rightarrow \dfrac{{2{{\cos }^{ - 1}}0.8}}{2} = \dfrac{\theta }{2} \\
   \Rightarrow {\cos ^{ - 1}}0.8 = \dfrac{\theta }{2} \\
 \]
Taking \[\cos \] on both sides in the above equation, we get
\[ \Rightarrow \cos \left( {{{\cos }^{ - 1}}0.8} \right) = \cos \dfrac{\theta }{2}\]
Using the inverse property of trigonometric functions, \[\cos \left( {{{\cos }^{ - 1}}x} \right) = x\] in the above equation, we get
\[
   \Rightarrow 0.8 = \cos \dfrac{\theta }{2} \\
   \Rightarrow \cos \dfrac{\theta }{2} = 0.8 \\
 \]
We know the property of trigonometric functions, \[1 + \cos \theta = 2{\cos ^2}\left( {\dfrac{\theta }{2}} \right)\].
Substituting the value of \[\cos \dfrac{\theta }{2}\] in the property of trigonometry, we get
\[
   \Rightarrow 1 + \cos \theta = 2{\left( {0.8} \right)^2} \\
   \Rightarrow 1 + \cos \theta = 2\left( {0.64} \right) \\
   \Rightarrow 1 + \cos \theta = 1.28 \\
 \]
Subtracting the above equation by 1 on both sides, we get
\[
   \Rightarrow 1 + \cos \theta - 1 = 1.28 - 1 \\
   \Rightarrow \cos \theta = 0.28 \\
 \]

Taking \[{\cos ^{ - 1}}\] on both sides and using the inverse property again in the above equation, we get
\[
   \Rightarrow {\cos ^{ - 1}}\left( {\cos \theta } \right) = {\cos ^{ - 1}}0.28 \\
   \Rightarrow \theta = {\cos ^{ - 1}}0.28 \\
 \]
Substituting the above value of \[\theta \] in the given equation, we get
\[
   \Rightarrow \cos \left( {2{{\cos }^{ - 1}}0.8} \right) \\
   = \cos \left( \theta \right) \\
   = \cos \left( {{{\cos }^{ - 1}}0.28} \right) \\
   = 0.28 \\
 \]
So, the answer is \[0.28\].
Hence, option D is correct.

Note In solving this question, we should know the basic properties of trigonometric functions, like \[1 + \cos \theta = 2{\cos ^2}\left( {\dfrac{\theta }{2}} \right)\] and \[\cos \left( {{{\cos }^{ - 1}}x} \right) = x\]. If students are familiar with the properties, then these types of questions are simple. Students have to be really careful while solving to avoid the calculation mistakes.