Geometry

# Advanced Pythagorean Identities

What is the value of

$\sin ^2 10 ^ \circ + \sin ^2 20 ^ \circ + \sin ^2 30 ^ \circ + \sin ^2 40 ^ \circ + \sin ^2 50 ^ \circ + \sin ^2 60 ^ \circ + \sin ^2 70 ^ \circ + \sin ^2 80 ^ \circ ?$

If $\sin \theta+ \cos \theta = \frac {1}{2}$, what is the value of $\sin \theta \cos \theta$?

If $\sin \theta+ \cos \theta = \frac {1}{2}$, what is the value of $\sin \theta-\cos \theta$?

If $\sin \theta+ \cos \theta = \frac {1}{2}$, what is the value of $\tan \theta+\cot \theta$?

If $\tan \theta+ \cot \theta = 3$, what is the value of $\sin \theta\cos \theta$?

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