Using the Addition Formulas


Use the addition formulas to derive the identities in Exercises 31–36.


cos (x − π/2) = sin x

Verified step by step guidance

1. Start by recalling the cosine addition formula: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
2. In this problem, set \( a = x \) and \( b = \frac{\pi}{2} \). Substitute these values into the addition formula: \( \cos(x - \frac{\pi}{2}) = \cos x \cos \frac{\pi}{2} + \sin x \sin \frac{\pi}{2} \).
3. Evaluate the trigonometric functions at \( \frac{\pi}{2} \): \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \).
4. Substitute these values back into the equation: \( \cos(x - \frac{\pi}{2}) = \cos x \cdot 0 + \sin x \cdot 1 \).
5. Simplify the expression: \( \cos(x - \frac{\pi}{2}) = \sin x \), which confirms the identity.

Key Concepts

Addition Formulas

Addition formulas are trigonometric identities that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, the cosine of the difference of two angles is given by cos(a - b) = cos(a)cos(b) + sin(a)sin(b). These formulas are essential for simplifying expressions and solving trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are used to simplify expressions and solve equations in trigonometry. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities, which relate different trigonometric functions to one another.

Co-function Identities

Co-function identities are specific trigonometric identities that relate the sine and cosine functions of complementary angles. For instance, sin(π/2 - x) = cos(x) and cos(π/2 - x) = sin(x). These identities are particularly useful when working with angles that involve π/2, as they allow for the transformation of sine and cosine functions into one another.