Verify that each equation is an identity.
(2 cot x)/(tan 2x) = csc² x - 2

Verified step by step guidance

1. Start by rewriting all trigonometric functions in terms of sine and cosine to simplify the expression. Recall that \(\cot x = \frac{\cos x}{\sin x}\) and \(\tan 2x = \frac{\sin 2x}{\cos 2x}\).
2. Substitute these into the left-hand side (LHS) of the equation: \(\frac{2 \cot x}{\tan 2x} = \frac{2 \frac{\cos x}{\sin x}}{\frac{\sin 2x}{\cos 2x}}\).
3. Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\frac{2 \frac{\cos x}{\sin x}}{\frac{\sin 2x}{\cos 2x}} = 2 \frac{\cos x}{\sin x} \times \frac{\cos 2x}{\sin 2x}\).
4. Use the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\), and substitute it into the expression to simplify further.
5. After simplification, compare the resulting expression to the right-hand side (RHS) \(\csc^2 x - 2\). Recall that \(\csc x = \frac{1}{\sin x}\) and \(\csc^2 x = 1 + \cot^2 x\) (Pythagorean identity). Use these identities to verify that both sides are equal, confirming the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means manipulating one side of the equation using known identities until it matches the other side, confirming their equivalence.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. For example, tan(2x) = 2 tan x / (1 - tan² x). These formulas are essential for simplifying expressions involving angles like 2x.

Reciprocal and Quotient Identities

Reciprocal identities relate functions like cot x = 1/tan x and csc x = 1/sin x, while quotient identities express functions like tan x = sin x / cos x. Using these helps rewrite expressions in simpler or more comparable forms to verify identities.