Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 66

Chapter 6, Problem 66

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

Verified step by step guidance

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.

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. 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.

Recommended video:

2:25

Verifying Identities with Sum and Difference Formulas

Related Practice

Textbook Question

Verify that each equation is an identity.

sin³ θ = sin θ - cos² θ sin θ

views

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

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.

cos 18°

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 102° - sin 95°

Textbook Question

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

cos( π/2 + x) = -sin x

views