Verify that each equation is an identity.
csc A sin 2A - sec A = cos 2A sec A

Verified step by step guidance

1. Start by writing down the given equation to verify: \(\csc A \sin 2A - \sec A = \cos 2A \sec A\).
2. Recall the double-angle identity for sine: \(\sin 2A = 2 \sin A \cos A\). Substitute this into the left-hand side (LHS) to get \(\csc A \cdot 2 \sin A \cos A - \sec A\).
3. Simplify the term \(\csc A \cdot \sin A\) by using the definition \(\csc A = \frac{1}{\sin A}\), so \(\csc A \sin A = 1\). This reduces the LHS to \(2 \cos A - \sec A\).
4. Express \(\sec A\) as \(\frac{1}{\cos A}\) and rewrite the LHS as \(2 \cos A - \frac{1}{\cos A}\). Find a common denominator to combine the terms into a single fraction.
5. Now, simplify the right-hand side (RHS) \(\cos 2A \sec A\) by writing \(\sec A = \frac{1}{\cos A}\), so RHS becomes \(\frac{\cos 2A}{\cos A}\). Use the double-angle identity for cosine, \(\cos 2A = 2 \cos^2 A - 1\), and compare the simplified forms of LHS and RHS to verify 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 showing both sides simplify to the same expression, often by applying known formulas and algebraic manipulation.

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as csc A = 1/sin A and sec A = 1/cos A. These are useful for rewriting expressions to simplify or transform terms when verifying identities.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2A in terms of functions of A, for example, sin 2A = 2 sin A cos A and cos 2A = cos² A - sin² A. These formulas help rewrite and simplify expressions involving angles multiplied by two.