Verify that each equation is an identity.
(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

Verified step by step guidance

1. Step 1: Use the sum-to-product identities for sine. The sum-to-product identities are: \(\sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\) and \(\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)\).
2. Step 2: Apply the sum-to-product identities to the numerator and denominator of the left-hand side. For the numerator \(\sin 3t + \sin 2t\), use \(A = 3t\) and \(B = 2t\). For the denominator \(\sin 3t - \sin 2t\), use the same values for \(A\) and \(B\).
3. Step 3: Simplify the expressions obtained from the sum-to-product identities. The numerator becomes \$2 \sin\left(\frac{5t}{2}\right) \cos\left(\frac{t}{2}\right)\( and the denominator becomes \)2 \cos\left(\frac{5t}{2}\right) \sin\left(\frac{t}{2}\right)$.
4. Step 4: Divide the simplified numerator by the simplified denominator. This results in \(\frac{2 \sin\left(\frac{5t}{2}\right) \cos\left(\frac{t}{2}\right)}{2 \cos\left(\frac{5t}{2}\right) \sin\left(\frac{t}{2}\right)}\).
5. Step 5: Simplify the expression by canceling out the common factor of 2 and using the identity \(\frac{\sin A}{\cos A} = \tan A\). This gives \(\tan\left(\frac{5t}{2}\right) / \tan\left(\frac{t}{2}\right)\), which matches the right-hand side of the original equation, verifying 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, angle sum and difference identities, and double angle formulas. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.

Sine and Tangent Functions

The sine function, sin(t), represents the ratio of the opposite side to the hypotenuse in a right triangle, while the tangent function, tan(t), is the ratio of the sine to the cosine of the angle, or sin(t)/cos(t). These functions are periodic and have specific properties that can be used to manipulate and verify trigonometric equations.

Algebraic Manipulation of Trigonometric Expressions

Algebraic manipulation involves rearranging and simplifying trigonometric expressions using identities and properties. This includes factoring, combining fractions, and substituting equivalent expressions. Mastery of these techniques is essential for verifying identities and solving trigonometric equations effectively.