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

Verified step by step guidance

Recall the angle addition formula for cosine: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\).

Apply the formula to the left side of the equation with \(a = \frac{\pi}{2}\) and \(b = x\): \(\cos\left(\frac{\pi}{2} + x\right) = \cos\frac{\pi}{2} \cos x - \sin\frac{\pi}{2} \sin x\).

Substitute the known values of \(\cos\frac{\pi}{2} = 0\) and \(\sin\frac{\pi}{2} = 1\) into the expression: \(0 \cdot \cos x - 1 \cdot \sin x\).

Simplify the expression to get \(-\sin x\) on the left side.

Since the right side of the original equation is \(-\sin x\), both sides are equal, verifying the identity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow the transformation and simplification of expressions, such as rewriting cos(π/2 + x) in terms of sine or cosine functions to verify equivalences.

Angle Addition Formulas

Angle addition formulas express trigonometric functions of sums of angles, like cos(a + b) = cos a cos b - sin a sin b. These formulas are essential for breaking down complex expressions, such as cos(π/2 + x), into simpler components to facilitate verification of identities.

Relationship Between Sine and Cosine

Sine and cosine functions are closely related, often differing by phase shifts. For example, cos(π/2 + x) equals -sin x, showing how a cosine function shifted by π/2 relates directly to sine. Understanding this relationship helps in recognizing and proving trigonometric identities.

Recommended video:

5:05

Amplitude and Reflection of Sine and Cosine

Related Practice

Textbook Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cot 18°

views

Textbook Question

Verify that each equation is an identity.

(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

Textbook Question

Verify that each equation is an identity.

2 cos² (x/2) tan x = tan x+ sin x

views

Textbook Question

Write each expression as a product of trigonometric functions. See Example 8.

sin 9x - sin 3x