Question

If $cos\theta =\frac{1}{2}(a+\frac{1}{a}),andcos3\theta =\lambda (a^3+\frac{1}{a^3}),then\lambda =$

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. 1
• D. none of these

Answer 1

The correct option is B

$\frac{1}{2}$

Given:

$cos\theta =\frac{1}{2}(a+\frac{1}{a})cos3\theta =\lambda (a^3+\frac{1}{a^3})$

Now,

$co{s}^3\theta =\frac{1}{8}[a^3+\frac{1}{a^3}+3a\frac{1}{a}(a+\frac{1}{a})]\Rightarrow co{s}^3\theta =\frac{1}{8}(a^3+\frac{1}{a^3}+3\times 2cos\theta )[\because cos\theta =\frac{1}{2}(a+\frac{1}{a})]\Rightarrow co{s}^3\theta =\frac{1}{8}(\frac{cos3\theta }{\lambda }+6cos\theta )\Rightarrow co{s}^3\theta =\frac{1}{8}(\frac{4co{s}^2\theta -3cos\theta }{\lambda }+6cos\theta )\Rightarrow co{s}^3\theta =\frac{4co{s}^2\theta }{8\lambda }-\frac{3cos\theta }{8\lambda }+\frac{6cos\theta }{8}$

On comparing the power of $co{s}^3\theta$ on both sides, we get $1=\frac{4}{4\lambda }$

$\Rightarrow \lambda =\frac{1}{2}$