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# Pythagorean Identities

Trigonometric identities bring new life to the Pythagorean theorem by re-envisioning the legs of a right triangle as sine and cosine.

Given that \(\cos{35°}=\alpha\), express \(\sin{2015°}\) in terms of \(\alpha\).

*This problem is part of the set 2015 Countdown Problems.*

\[\dfrac1{\sin^2{\theta}}-\dfrac1{\cos^2{\theta}}-\dfrac1{\tan^2{\theta}}-\dfrac1{\cot^2{\theta}}-\dfrac1{\sec^2{\theta}}-\dfrac1{\csc^2{\theta}}=-3\]

Find the number of solutions of \(\theta\) in the interval \((0,2\pi)\) that satisfy the equation above.

\[ \large 4\sin^2 (x) + \tan^2(x) + \cot^2(x) + \csc^2(x) = 6 \]

Find the number of solutions of \(x\) in the interval \([0,2\pi]\) that satisfy the equation above.

As \(x \) ranges over all real values, what is the range of

\[ A=\sin^4x+\cos^2x?\]

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