Unit 5 · Topic 3.12

Equivalent Representations of Trigonometric Functions

Rewrite trigonometric expressions into forms that reveal values, domains, graph features, or equation solutions more clearly.

Learning Goals

Derive and rearrange the Pythagorean identities.
Use reciprocal, quotient, and symmetry relationships with domain awareness.
Apply sine and cosine sum, difference, and double-angle identities.
Verify identities through valid algebraic steps.
Select an equivalent form that makes a value or equation easier to determine.

1. Equivalent Forms Serve Different Purposes

An identity states that two expressions have equal outputs for every input where both sides are defined. Equivalent forms may expose different information.

\(1-\sin^2x\) highlights subtraction from 1.
\(\cos^2x\) is simpler when solving for cosine.
\((1-\cos x)(1+\cos x)\) reveals a difference-of-squares factorization.

Choose the form that makes the next mathematical step visible.

2. Derive the Fundamental Pythagorean Identity

A point on the unit circle has coordinates \((\cos\theta,\sin\theta)\). Substituting these coordinates into \(x^2+y^2=1\) gives

\[\cos^2\theta+\sin^2\theta=1.\]

The exponent applies to the function value: \(\sin^2\theta\) means \((\sin\theta)^2\), not \(\sin(\theta^2)\).

3. Derive Two More Pythagorean Identities

Divide \(\sin^2x+\cos^2x=1\) by \(\cos^2x\), where \(\cos x\ne0\):

\[\tan^2x+1=\sec^2x.\]

Divide instead by \(\sin^2x\), where \(\sin x\ne0\):

\[1+\cot^2x=\csc^2x.\]

The restrictions come from the division used in the derivation and match the domains of the resulting reciprocal functions.

4. Rearrange before Substituting

Known expression	Useful equivalent form
\(1-\sin^2x\)	\(\cos^2x\)
\(1-\cos^2x\)	\(\sin^2x\)
\(\sec^2x-1\)	\(\tan^2x\)
\(\csc^2x-1\)	\(\cot^2x\)

For example, \(\sqrt{1-\sin^2x}=|\cos x|\), not always \(\cos x\). The square root returns a nonnegative value.

5. Compare Equivalent Graphs

Equivalent formulas overlap exactly wherever both are defined.

Graphing two equivalent expressions over their common domain produces overlapping curves, so only one curve may appear. A graph supports an identity, but algebra and domain analysis provide the justification.

6. Recall Reciprocal, Quotient, and Symmetry Forms

Relationship type	Examples
Reciprocal	\(\sec x=1/\cos x,\ \csc x=1/\sin x\)
Quotient	\(\tan x=\sin x/\cos x,\ \cot x=\cos x/\sin x\)
Even	\(\cos(-x)=\cos x,\ \sec(-x)=\sec x\)
Odd	\(\sin(-x)=-\sin x,\ \tan(-x)=-\tan x\)

Converting everything to sine and cosine is often useful when an expression contains several different trigonometric functions.

7. Sine and Cosine Sum and Difference Identities

Function	Sum	Difference
Sine	\(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\)	\(\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\)
Cosine	\(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\)	\(\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\)

Sine keeps the sign between the angles. Cosine uses the opposite sign between its two products.

8. Avoid the Distribution Trap

Trigonometric functions do not distribute over addition:

\[\sin(\alpha+\beta)\ne\sin\alpha+\sin\beta,\]

\[\cos(\alpha+\beta)\ne\cos\alpha+\cos\beta.\]

For a quick counterexample, let \(\alpha=\beta=\pi/2\). Then \(\sin(\alpha+\beta)=0\), while \(\sin\alpha+\sin\beta=2\).

9. Find an Exact Value with a Sum

Rewrite \(75^\circ=45^\circ+30^\circ\):

\[\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ.\]

\[\sin75^\circ=\frac{\sqrt2}{2}\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\frac12=\frac{\sqrt6+\sqrt2}{4}.\]

The positive result agrees with a Quadrant I angle.

10. Find an Exact Value with a Difference

Rewrite \(105^\circ=135^\circ-30^\circ\):

\[\cos105^\circ=\cos135^\circ\cos30^\circ+\sin135^\circ\sin30^\circ.\]

\[\cos105^\circ=-\frac{\sqrt2}{2}\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\frac12=\frac{\sqrt2-\sqrt6}{4}.\]

The expression is negative, as expected because \(105^\circ\) lies in Quadrant II.

11. Derive the Double-Angle Identities

Set \(\alpha=\beta=x\) in the sum identities:

\[\sin(2x)=2\sin x\cos x,\]

\[\cos(2x)=\cos^2x-\sin^2x.\]

Use \(\sin^2x+\cos^2x=1\) to obtain two more cosine forms:

\[\cos(2x)=2\cos^2x-1=1-2\sin^2x.\]

Select the version containing the function already present in the problem.

12. Verify an Identity Algebraically

Verify \(\sec x-\cos x=\sin x\tan x\) where \(\cos x\ne0\). Start with the more complex left side:

\[\sec x-\cos x=\frac1{\cos x}-\cos x=\frac{1-\cos^2x}{\cos x}.\]

\[=\frac{\sin^2x}{\cos x}=\sin x\left(\frac{\sin x}{\cos x}\right)=\sin x\tan x.\]

Every step preserves equality on the common domain. This is verification, not solving for selected values of \(x\).

13. Solve with a More Useful Representation

Solve \(\cos(2x)=\cos x\) on \(0\le x<2\pi\). Rewrite the double angle using only cosine:

\[2\cos^2x-1=\cos x.\]

\[(2\cos x+1)(\cos x-1)=0.\]

Thus \(\cos x=-1/2\) or \(\cos x=1\), giving

\[x=0,\quad\frac{2\pi}{3},\quad\frac{4\pi}{3}.\]

14. Domain Restrictions Can Change a Claim

Let \(0\le x\le1\) and \(\theta=\arcsin x\). Then \(\theta\in[0,\pi/2]\), so \(\cos\theta\ge0\) and

\[\cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-x^2}.\]

\[\arcsin x=\arccos\left(\sqrt{1-x^2}\right),\qquad 0\le x\le1.\]

For negative \(x\), the left side is negative while the right side is nonnegative, so the restriction is essential.

15. AP Workflow and Common Errors

Identify the target function or form.
Start with the more complicated expression.
Factor, combine fractions, or convert to sine and cosine.
Apply one justified identity at a time.
Track every denominator and square-root restriction.
Use the new form to evaluate, verify, or solve.
Check the result numerically or graphically without treating that check as proof.

Do not distribute sine or cosine across a sum.
Do not change the sign pattern in the sum identities.
Do not replace \(\sqrt{\cos^2x}\) with \(\cos x\) instead of \(|\cos x|\).
Do not divide by an expression that may be zero without considering that case.
Do not prove an identity by assuming the statement to be proved.

Key Takeaways

Use identities to rewrite, solve, and verify equivalent trigonometric expressions.
Core relationship: \(\sin^2x+\cos^2x=1,\quad1+\tan^2x=\sec^2x\)
Error check: Verify domains before claiming two reciprocal-form expressions are equivalent.

Checkpoint · Topic 3.12

Rewrite \(1-\cos^2x\) and \(\sec^2x-1\).
Find the exact value of \(\cos15^\circ\).
Derive \(\cos(2x)=1-2\sin^2x\) from the cosine sum identity.
Verify \((1-\sin x)(1+\sin x)=\cos^2x\).
Solve \(\sin(2x)=\sin x\) on \(0\le x<2\pi\).