Verify each identity. csc θ - sin θ = cot θ cos θ

Verified step by step guidance

1. Start by writing down the given identity to verify: \(\csc \theta - \sin \theta = \cot \theta \cos \theta\).
2. Recall the fundamental trigonometric definitions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
3. Rewrite the left-hand side (LHS) using the definition of cosecant: \(\csc \theta - \sin \theta = \frac{1}{\sin \theta} - \sin \theta\).
4. Find a common denominator on the LHS to combine the terms: \(\frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}\).
5. Use the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\) to simplify the numerator, so the LHS becomes \(\frac{\cos^2 \theta}{\sin \theta}\). Then compare this with the right-hand side (RHS) \(\cot \theta \cos \theta = \frac{\cos \theta}{\sin \theta} \times \cos \theta = \frac{\cos^2 \theta}{\sin \theta}\) 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 of the equation are equivalent by using known identities and algebraic manipulation.

Reciprocal and Quotient Identities

Reciprocal identities relate sine, cosine, and tangent to their reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot). For example, csc θ = 1/sin θ and cot θ = cos θ/sin θ. These are essential for rewriting expressions to verify identities.

Algebraic Manipulation of Trigonometric Expressions

Algebraic manipulation involves factoring, combining fractions, and simplifying expressions using trigonometric identities. This skill is crucial to transform one side of the identity into the other, often by expressing all terms in sine and cosine.