Given sin^10(x) < or = 1 - solve this equation : sin^10(x) - cos^10(x) = 1

Question

How to calculate that sin x to the power 10 minus cosine x to the power 10 equals 1??

Answer 1

When x=90°, sin(x)=1 and sin¹⁰(x)=1, cos(x)=cos¹⁰(x)=0.

When x=270°, sin(x)=-1 and sin¹⁰(x)=1, cos(x)=0.

So sin¹⁰(x)-cos¹⁰(x)=1 when x=90° or 270° or, more generally, x=90(2n+1)° where n is an integer.

When x=0, 180°, or, more generally, x=180m where m is an integer, then sin(x)=0 and cos(x)=1 or -1, so cos¹⁰(x)=1 and sin¹⁰(x)-cos¹⁰(x)=-1, which doesn’t satisfy the given equation.

We can also let y=sin²(x) then cos²(x)=1-y.

So the equation becomes y⁵-(1-y)⁵=1 where 0≤y≤1. From this, y⁵=1+(1-y)⁵. The expression on the right exceeds 1 unless y=1 which balances the equation. We know by this that there are no more solutions so sin(x)=±1 and x=90(2n+1)° where n is an integer.