If-270-lt-x-lt-360-simplify-2-2-2cosx-

Question Number 8306 by lepan last updated on 07/Oct/16

I f 270 ° < x < 360 °, s i m p l i f y

\sqrt{2 + \sqrt{2 + 2 c o s x}} .

Commented by sou1618 last updated on 07/Oct/16

\sqrt{2 + \sqrt{2 + 2 c o s x}} (270 ° < x < 360 °)

2 {c o s}^{2} α = 1 + c o s 2 α

= \sqrt{2 + \sqrt{4 {c o s}^{2} (\frac{1}{2} x)}}

= \sqrt{2 + ∣ 2 c o s (\frac{1}{2} x) ∣}

135 ° < \frac{1}{2} x < 180 ° \Rightarrow c o s (\frac{1}{2} x) < 0

= \sqrt{2 - 2 c o s (\frac{1}{2} x)}

2 {s i n}^{2} α = 1 - c o s 2 α

= \sqrt{4 {s i n}^{2} (\frac{1}{4} x)}

=∣ 2 s i n (\frac{1}{4} x) ∣

67.5 ° < \frac{1}{4} x < 90 ° \Rightarrow s i n (\frac{1}{4} x) > 0

= 2 s i n \frac{1}{4} x

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