1.

Find the value of \(\sqrt {2\; + \;\sqrt {2\; + \;\sqrt {2\; + \;2{\rm{cos}}16{\rm{\theta }}} \;} \;} \)1. 2cosθ2. 2cos2θ3. cos4θ4. 2cos(θ/2)

Answer» Correct Answer - Option 2 : 2cos2θ

Given:

\(\sqrt {2\; + \;\sqrt {2\; + \;\sqrt {2\; + \;2{\rm{cos}}16{\rm{\theta }}} \;} \;} \)

Concept used:

\(\sqrt {2\; + \;\sqrt {2\; + \;\sqrt {2\; + \;2{\rm{cos}}16{\rm{\theta }}} \;} \;} = \;2{\rm{cos}}\frac{{16\theta }}{{{2^{\rm{n}}}}}\)

Where n → number of terms

Calculation:

Here 2 came 3 times

⇒ n = 3

\(\sqrt {2\; + \;\sqrt {2\; + \;\sqrt {2\; + \;2{\rm{cos}}16{\rm{\theta }}} \;} \;} = \;2{\rm{cos}}\frac{{16\theta }}{{{2^3}}}\)

⇒ 2cos2θ

\(\therefore {\bf{The}}\;{\bf{value}}\;{\bf{of}}\;\sqrt {2\; + \;\sqrt {2\; + \;\sqrt {2\; + \;2{\bf{cos}}16{\bf{\theta }}} \;} \;} {\bf{is}}\;2{\bf{cos}}2{\bf{\theta }}.\)


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