Verify that each equation is an identity.
sin(s + t)/cos s cot t = tan s + tan t

검증된 단계별 안내

Start by writing down the given equation to verify: \(\frac{\sin(s + t)}{\cos s} \cot t = \tan s + \tan t\).

Recall the angle sum identity for sine: \(\sin(s + t) = \sin s \cos t + \cos s \sin t\). Substitute this into the left-hand side (LHS) of the equation.

Rewrite \(\cot t\) as \(\frac{\cos t}{\sin t}\) and substitute it into the LHS, so the expression becomes \(\frac{\sin s \cos t + \cos s \sin t}{\cos s} \times \frac{\cos t}{\sin t}\).

Simplify the LHS by distributing and canceling terms where possible. Break the fraction into two parts: \(\frac{\sin s \cos t}{\cos s} \times \frac{\cos t}{\sin t} + \frac{\cos s \sin t}{\cos s} \times \frac{\cos t}{\sin t}\).

Express the resulting terms in terms of tangent functions, using \(\tan x = \frac{\sin x}{\cos x}\), and simplify to show that the LHS equals the right-hand side (RHS), \(\tan s + \tan t\), thus verifying the identity.

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주요 개념

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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, reciprocal, or quotient identities.

Quotient and Reciprocal Identities

Quotient identities express tangent and cotangent in terms of sine and cosine: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Reciprocal identities relate sine, cosine, and their reciprocals cosecant and secant. These are essential for rewriting expressions to simplify or prove identities.

Sum of Angles Formulas

Sum of angles formulas provide expressions for trigonometric functions of sums, such as sin(s + t) = sin s cos t + cos s sin t. These formulas are crucial for expanding or simplifying expressions involving sums of angles to verify identities.

Verify that each equation is an identity.sin(s + t)/cos s cot t = tan s + tan t