Simplify the expression.
$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)\csc^2\left(\theta\right)\cos^2\left(-\theta\right)$

Solving Trigonometric Equations

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.

sin² θ = cos² θ

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

Simplify the expression.

$\tan^2\theta-\sec^2\theta+1$

Simplify the expression.

$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

Identify the most helpful first step in verifying the identity.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)=\sec^2\theta\sin^2\left(-\theta\right)$

Identify the most helpful first step in verifying the identity.

$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

Find all solutions to the equation.

$\left(\cos\theta+\sin\theta\right)\left(\cos\theta-\sin\theta\right)=-\frac12$

Find all solutions to the equation where 0 ≤ $\theta$ ≤ $2\pi$.

$\sin\theta\cos\left(2\theta\right)-\sin\left(2\theta\right)\cos\theta=\frac{\sqrt2}{2}$