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

Verified step by step guidance

1. Start by recalling the half-angle identity for tangent: \(\tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta}\) or alternatively \(\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta}\). Choose the form that seems easier to work with for this problem.
2. Rewrite the right-hand side expression \(\csc \theta - \cot \theta\) in terms of sine and cosine functions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). So, \(\csc \theta - \cot \theta = \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}\).
3. Combine the terms on the right-hand side over a common denominator \(\sin \theta\): \(\frac{1 - \cos \theta}{\sin \theta}\).
4. Compare this simplified right-hand side expression \(\frac{1 - \cos \theta}{\sin \theta}\) with the half-angle identity form of \(\tan\left(\frac{\theta}{2}\right)\) you selected in step 1. They should match, confirming the identity.
5. Conclude that since both sides simplify to the same expression, the given equation \(\tan\left(\frac{\theta}{2}\right) = \csc \theta - \cot \theta\) is indeed an 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 using fundamental identities like Pythagorean or reciprocal identities.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, tan(θ/2) can be written using sine and cosine of θ, which helps in transforming and simplifying expressions involving half angles.

Reciprocal and Quotient Identities

Reciprocal identities relate functions like cosecant (csc θ = 1/sin θ) and cotangent (cot θ = cos θ/sin θ) to sine and cosine. Quotient identities express tangent and cotangent as ratios of sine and cosine, facilitating the manipulation and verification of trigonometric equations.