Question
Simplify the square roots expression
\sqrt{2+\sqrt{2+\sqrt{3} } } - \sqrt{3} \sqrt{2-\sqrt{2+\sqrt{3} } }
Simplify the square roots expression
\sqrt{2+\sqrt{2+\sqrt{3} } } - \sqrt{3} \sqrt{2-\sqrt{2+\sqrt{3} } }
The special values in
reminds us of the trigonometric values of special angles.
From our previous posts, we have derived the exact values for \sin 15° and \cos 15°,
We are going to use the value of the special angle to simplify the square root expression.
Rearranging (3) gives
Substitute (4) to (1), we get
Then, using half angle identities
gives
Substitute them to (5), we get
\begin{aligned} 2 \cos 7.5° - 2\sqrt{3} \sin 7.5° &=4(\dfrac{1}{2} \cos 7.5° - \dfrac{\sqrt{3} }{2} \sin 7.5°) \\&=4(\sin 30° \cos 7.5° - \cos 30° \sin 7.5°) \\&= 4\sin(30°-7.5° )\\&=4\sin 22.5° \end{aligned}
In summary, the square root express
is simplified as