Question Type 6: Applying the golden rule to rewrite values of trigonometric functions Exercises

Simplify the ratio $\frac{5 - 5\sin^2 x}{5 - 5\cos^2 x}$.

Simplify the expression $5 - 5\cos^2\left(x + \frac{\pi}{4}\right)$. [2 marks]

Express $5 - 5\sin^2(2x)$ in terms of $\cos(4x)$.

Express $\sqrt{5 - 5\sin^2 x}$ in terms of $\cos x$, indicating any absolute values required.

Evaluate the indefinite integral $\displaystyle \int (5 - 5\sin^2 x)\,dx$.

Solve the equation $\sqrt{5 - 5\sin^2 x} = 3$ for $x$ in $[0,2\pi)$.

Differentiate $f(x) = \sqrt{5 - 5\sin^2 x}$ with respect to $x$.

In a right triangle, the hypotenuse has length $\sqrt{5}$ and the side opposite angle $x$ has length $\sqrt{5}\sin x$. Show that the length of the adjacent side is $\sqrt{5 - 5\sin^2 x}$ and then simplify it.

Solve $5 - 5\sin^2 x = \cos x$ for $x$ in $[0,2\pi)$.

Simplify the square of the sum $(\sin x + \sqrt{5 - 5\sin^2 x})^2$.

Simplify the expression $5 - 5\sin^2 x$ using a Pythagorean identity.