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.