Paramteric Curves and the exponents of $\cos$/$\sin$/$\tan$

Question

Lets say we have the curve $\frac x7=\cos^7t$, $\frac y7=\sin^7t$

Now I know that $\sin^2x+\cos^2x=1$.

So $\cos^2=(\frac x7)^{\text{some exponent}}$.

What is that exponent? How do you work it out?

Answer 1

I think this problem wouldn't be giving you trouble if you knew some properties of exponents. First of all, $f(x) = x^n$ is a function for all n. This means one can raise both sides of an equation to a power and the equality will hold. Another property is that $(a^b)^c = a^{bc}$. Using these properties, you don't even need the trigonometric pythagorean identity, just manipulate the equation $\frac x7 = \cos^7t$.