Verify that each equation is an identity.
sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x

Verified step by step guidance

1. Start by recalling the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\).
2. Express \(\cot x\) and \(\tan x\) in terms of \(\sin x\) and \(\cos x\): \(\cot x = \frac{\cos x}{\sin x}\) and \(\tan x = \frac{\sin x}{\cos x}\).
3. Substitute \(\cot x\) and \(\tan x\) into the equation: \(\sin^2 x(1 + \frac{\cos x}{\sin x}) + \cos^2 x(1 - \frac{\sin x}{\cos x}) + \cot^2 x\).
4. Simplify each term: \(\sin^2 x + \sin x \cos x + \cos^2 x - \sin x \cos x + \cot^2 x\).
5. Use the identity \(\cot^2 x = \csc^2 x - 1\) to simplify further and verify the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.

Reciprocal Functions

Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). For example, csc x = 1/sin x and cot x = 1/tan x. Recognizing these relationships helps in transforming and simplifying trigonometric expressions.

Simplification Techniques

Simplification techniques involve manipulating trigonometric expressions using identities to make them easier to analyze or verify. This can include factoring, combining like terms, or substituting equivalent expressions. Mastery of these techniques is essential for proving that an equation is an identity.