6.1: Fundamental Identities | Precalculus

How it works

An identity holds for all inputs where both sides are defined. Rewrite using sine and cosine, use the Pythagorean identity, and keep original domain exclusions even when an expression simplifies.

\sin^2x+\cos^2x=1,\quad1+\tan^2x=\sec^2x

Worked examples

Example 1

Simplify $\displaystyle 1-\cos^2x$ .

The Pythagorean identity gives $\displaystyle \sin^2x$ .

Example 2

Simplify $\displaystyle (1-\cos^2x)/\sin x$ and state its restriction.

The numerator is $\displaystyle \sin^2x$ , so the expression equals $\displaystyle \sin x$ where $\displaystyle \sin x\ne0$ .

Problems

Problem 6.1.1

Simplify $\displaystyle \sec^2x-\tan^2x$ .

Show worked solution

The identity $\displaystyle 1+\tan^2x=\sec^2x$ gives $\displaystyle 1$ , with $\displaystyle \cos x\ne0$ retained.

Problem 6.1.2

If $\displaystyle \sin x=3/5$ and x is in quadrant II, find cosine.

Show worked solution

$\displaystyle \cos^2x=1-9/25=16/25$ . Quadrant II gives $\displaystyle \cos x=-4/5$ .

Problem 6.1.3

Show that $\displaystyle \tan x\cos x=\sin x$ and state the original restriction.

Show worked solution

Replace tangent by $\displaystyle \sin x/\cos x$ and cancel. The original expression requires $\displaystyle \cos x\ne0$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 6.1.1

Problem 6.1.2

Problem 6.1.3

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