Verify each identity. cos θ sec θ/cot θ= tan θ

Verified step by step guidance

1. Start by writing down the given expression clearly: \(\frac{\cos \theta \cdot \sec \theta}{\cot \theta} = \tan \theta\).
2. Recall the fundamental trigonometric identities: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
3. Substitute these identities into the expression to rewrite it in terms of sine and cosine: \(\frac{\cos \theta \cdot \frac{1}{\cos \theta}}{\frac{\cos \theta}{\sin \theta}}\).
4. Simplify the numerator first: \(\cos \theta \cdot \frac{1}{\cos \theta} = 1\), so the expression becomes \(\frac{1}{\frac{\cos \theta}{\sin \theta}}\).
5. Simplify the complex fraction by multiplying numerator and denominator: \(\frac{1}{\frac{\cos \theta}{\sin \theta}} = \frac{\sin \theta}{\cos \theta}\), which is exactly \(\tan \theta\). This verifies 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 that both sides simplify to the same expression using known identities, such as reciprocal, quotient, or Pythagorean identities.

Reciprocal and Quotient Identities

Reciprocal identities relate functions like sec θ = 1/cos θ and cot θ = 1/tan θ. Quotient identities express tan θ as sin θ/cos θ and cot θ as cos θ/sin θ. These relationships help rewrite expressions to simplify and verify identities.

Simplification Techniques in Trigonometry

Simplification involves rewriting expressions using algebraic manipulation and trigonometric identities to reduce complex fractions or products. Recognizing common factors and converting all terms to sine and cosine often makes verification straightforward.