If 0 180 , then 2 + 2 + 2 + ? + 2 ( 1 + cos ) there being n number of 2's, is equal to

Question

If $0 ° < θ < 180 °$ , then $\sqrt{2 + \sqrt{2 + \sqrt{2 + \dots + \sqrt{2 (1 + \cos θ)}}}}$ there being $n$ number of 2's, is equal to

Smart Achievers · Accepted Answer

(1)

$\sqrt{2 + \sqrt{2 + \sqrt{2 + \dots + \sqrt{2 (1 + \cos θ)}}}},$ there being $n$ numbers of 2's

$= \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots + \sqrt{2 (2 \cos^{2} \frac{θ}{2})}}}}$

$= \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots + \sqrt{2 (1 + \cos \frac{θ}{2})}}}},$

there being $(n - 1)$ number of 2's,

$= \dots = \sqrt{2 (1 + \cos \frac{θ}{2^{n - 1}})} = 2 \cos \frac{θ}{2^{n}}$

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