Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos( π/2 + x) = -sin x

Verified step by step guidance

1. Start by using the angle addition formula for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
2. Apply the formula to \( \cos(\pi/2 + x) \), where \( a = \pi/2 \) and \( b = x \).
3. Substitute \( \cos(\pi/2) = 0 \) and \( \sin(\pi/2) = 1 \) into the formula: \( \cos(\pi/2 + x) = \cos(\pi/2)\cos x - \sin(\pi/2)\sin x \).
4. Simplify the expression: \( \cos(\pi/2 + x) = 0 \cdot \cos x - 1 \cdot \sin x \).
5. Conclude that \( \cos(\pi/2 + x) = -\sin x \), 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. They are fundamental in simplifying expressions and solving trigonometric equations. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas, which are essential for verifying equations as identities.

Angle Sum Identity for Cosine

The angle sum identity for cosine states that cos(a + b) = cos(a)cos(b) - sin(a)sin(b). This identity is crucial for expanding expressions involving the sum of angles, such as cos(π/2 + x). Understanding this identity allows one to rewrite the left-hand side of the equation in terms of sine and cosine functions, facilitating verification of the identity.

Co-function Identities

Co-function identities relate the trigonometric functions of complementary angles. For example, sin(π/2 - x) = cos(x) and cos(π/2 - x) = sin(x). These identities are useful in transforming trigonometric expressions and can help in proving identities by substituting one function for another, particularly when dealing with angles like π/2.