Trigonometric Identities Made Simple: Pythagorean Identities and Their Variants

Deriving sin2(x) + cos2(x) = 1 from the Unit Circle

On the unit circle, every point has coordinates (cos(x), sin(x)). The phrase “unit circle” means the radius is 1, so every point on the circle is exactly 1 unit from the origin (0,0).

Using the distance formula from the origin to a point (x,y), we have x2 + y2 = 1 for any point on the unit circle.

Substitute x = cos(x) and y = sin(x):

cos2(x) + sin2(x) = 1

Reordering terms gives the most common form:

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sin2(x) + cos2(x) = 1

Why this identity is so useful

This identity lets you replace a sin2 term with 1 − cos2, or replace a cos2 term with 1 − sin2. It is also the starting point for the other Pythagorean identities.

Deriving 1 + tan2(x) = sec2(x)

Start with the unit-circle identity:

sin2(x) + cos2(x) = 1

Divide every term by cos2(x) (this is valid wherever cos(x) ≠ 0):

sin2(x)/cos2(x) + cos2(x)/cos2(x) = 1/cos2(x)

Simplify each fraction:

• sin2(x)/cos2(x) = (sin(x)/cos(x))2 = tan2(x)
• cos2(x)/cos2(x) = 1
• 1/cos2(x) = (1/cos(x))2 = sec2(x)

So:

tan2(x) + 1 = sec2(x)

Usually written as:

1 + tan2(x) = sec2(x)

Deriving 1 + cot2(x) = csc2(x)

Again start with:

sin2(x) + cos2(x) = 1

Divide every term by sin2(x) (valid wherever sin(x) ≠ 0):

sin2(x)/sin2(x) + cos2(x)/sin2(x) = 1/sin2(x)

Simplify:

• sin2(x)/sin2(x) = 1
• cos2(x)/sin2(x) = (cos(x)/sin(x))2 = cot2(x)
• 1/sin2(x) = (1/sin(x))2 = csc2(x)

So:

1 + cot2(x) = csc2(x)

A Decision Guide: Choosing the Best Pythagorean Form

When simplifying or rewriting expressions, the main skill is choosing the identity form that reduces complexity the fastest.

Micro-strategy: pick the side with fewer terms

If your expression contains 1 + tan2(x), replacing it with sec2(x) reduces two terms to one. If your expression contains sec2(x) but you need a tan2(x) to combine with something else, rewrite sec2(x) as 1 + tan2(x).

Worked Examples (Step-by-Step): Choosing the Fastest Simplification

Example 1: Collapse a sum of squares

Simplify: 5(sin2(x) + cos2(x)) − 3

Step 1: Recognize sin2 + cos2 and aim for 1.

Step 2: Substitute sin2(x) + cos2(x) = 1.

5(1) − 3 = 2

Example 2: Use the tan–sec identity to reduce terms

Simplify: sec2(x) − tan2(x)

Step 1: Recognize it matches a rearranged Pythagorean identity.

From 1 + tan2(x) = sec2(x), subtract tan2(x) from both sides:

1 = sec2(x) − tan2(x)

Answer: 1

Example 3: Decide whether to rewrite sec2 or 1+tan2

Simplify: sec2(x) + tan2(x)

Decision: There is no direct identity for “plus.” But rewriting sec2(x) as 1 + tan2(x) allows combining like terms.

Step 1: Substitute sec2(x) = 1 + tan2(x).

sec2(x) + tan2(x) = (1 + tan2(x)) + tan2(x)

Step 2: Combine:

1 + 2tan2(x)

Example 4: Use the cot–csc identity in a subtraction pattern

Simplify: csc2(x) − cot2(x)

Step 1: Recognize the pattern csc2 − cot2.

From 1 + cot2(x) = csc2(x), subtract cot2(x):

1 = csc2(x) − cot2(x)

Answer: 1

Example 5: Choose which square to eliminate

Simplify: 1 − sin2(x)

Decision: This is almost the unit-circle identity solved for cos2(x).

Step 1: From sin2(x) + cos2(x) = 1, subtract sin2(x):

cos2(x) = 1 − sin2(x)

Answer: cos2(x)

Practice Set A: Spot the Identity and Simplify Quickly

Directions: Simplify each expression as much as possible by choosing the best Pythagorean identity form.

• 1) 7(sin2(x) + cos2(x))
• 2) sec2(x) − 1
• 3) 1 + tan2(x)
• 4) csc2(x) − cot2(x)
• 5) 1 − cos2(x)
• 6) (1 + tan2(x)) − sec2(x)

Practice Set B: Choose the Form That Reduces Complexity Fastest

Directions: For each, decide which substitution is most efficient, then simplify.

• 1) tan2(x) + 1 − sec2(x)
• 2) 2sec2(x) − 2tan2(x)
• 3) 3 − 3sin2(x)
• 4) 4csc2(x) − 4
• 5) cot2(x) + csc2(x)

Worked Solutions for Selected Practice Problems

Set A, Problem 2

Simplify: sec2(x) − 1

Decision: This resembles 1 + tan2(x) = sec2(x). Solve for tan2(x) by subtracting 1.

Step 1: sec2(x) = 1 + tan2(x)

Step 2: Subtract 1: sec2(x) − 1 = tan2(x)

Answer: tan2(x)

Set A, Problem 6

Simplify: (1 + tan2(x)) − sec2(x)

Decision: The expression literally contains both sides of the identity 1 + tan2(x) = sec2(x).

Step 1: Replace 1 + tan2(x) with sec2(x).

(1 + tan2(x)) − sec2(x) = sec2(x) − sec2(x)

Answer: 0

Set B, Problem 2

Simplify: 2sec2(x) − 2tan2(x)

Decision: Factor out 2, then use sec2 − tan2 = 1.

Step 1: Factor:

2(sec2(x) − tan2(x))

Step 2: Substitute sec2(x) − tan2(x) = 1.

2(1) = 2

Set B, Problem 5

Simplify: cot2(x) + csc2(x)

Decision: There is no direct identity for “plus,” but rewriting csc2(x) as 1 + cot2(x) creates like terms.

Step 1: Substitute csc2(x) = 1 + cot2(x).

cot2(x) + csc2(x) = cot2(x) + (1 + cot2(x))

Step 2: Combine:

1 + 2cot2(x)