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

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

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

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

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.

Simplify the expression $5 - 5\\cos^2\\bigl(x + \\tfrac{\\pi}{4}\\bigr)$.

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

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

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

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

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

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