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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.

# Pythagorean Identities: Level 3 Challenges

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.

For real number $$\theta$$ satisfying $$1 + 2 \sin^2{\theta} = 75 \, \cos^3{\theta}$$, what is the value of $$3 + 4 \tan^4{\theta}$$?

As $$x$$ ranges over all real values, what is the range of

$A=\sin^4x+\cos^2x?$

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