Inspired by soviet problem

Geometry Level 4

Find the sum of the solutions, IN DEGREES, to the equation sin ⁡ 2015 x − cos ⁡ 2015 x = 1 cos ⁡ x − 1 sin ⁡ x \displaystyle \sin^{2015} x - \cos^{2015} x = \frac{1}{\cos x} - \frac{1}{\sin x} from [ 0 ∘ , 36 0 ∘ ] [0^{\circ},360^{\circ}] .


The answer is 270.

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Hobart Pao by Hobart Pao , 23, USA

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1 solution

Hobart Pao
Apr 28, 2015

What I noticed is that if you replace 2015 with 3 or 5 or any odd number, when you factor, you're always going to have (sin x - cos x ) as a factor when the equation is set equal to zero, which gives 45 degrees and 225 degrees as solutions. When you factor, use factoring and pythagorean identities to reduce the equation into smaller bits until you end up with (sin x- cos x) multiplied by some other stuff, set equal to zero.

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