Questions: Pythagorean Trigonometric Identities

Question 1

Which of the following correctly simplifies the expression (1 − sin²x) / cos(x)?

• A. 1 / cos(x)
• B. cos(x)
• C. sin(x)
• D. tan(x)

Answer: (B)

The numerator 1 − sin²x is a rearrangement of the Pythagorean identity sin²x + cos²x = 1, giving cos²x. So the expression becomes cos²x / cos(x) = cos(x). The key skill being tested is recognizing that '1 − sin²x' is not a new expression — it equals cos²x by the identity. Students who don't recognize this form get stuck treating the numerator as irreducible.

Question 2

A student sees sec²(x) − 1 in an expression and needs to simplify it. Which identity applies most directly?

• A. sin²(x) + cos²(x) = 1
• B. 1 + cot²(x) = csc²(x)
• C. tan²(x) + 1 = sec²(x), rearranged as sec²(x) − 1 = tan²(x)
• D. cos²(x) = 1 − sin²(x)

Answer: (C)

Rearranging tan²x + 1 = sec²x gives sec²x − 1 = tan²x directly. This is a 'disguised form' — the same Pythagorean identity written with terms moved to the other side. Recognizing sec²x − 1 as tan²x is the kind of two-directional fluency the identities require. Students who only know the identity in one canonical form (tan² + 1 = sec²) but not its rearrangements will miss applications like this.

Question 3

The identity 1 + tan²(x) = sec²(x) is an independent mathematical fact that should be memorized separately from sin²(x) + cos²(x) = 1.

• T. True
• F. False

Answer: False

1 + tan²x = sec²x follows directly from dividing every term of sin²x + cos²x = 1 by cos²x. It is the same identity in a different form, not an independent fact. The same is true of 1 + cot²x = csc²x (obtained by dividing by sin²x). If you forget the derived identities on an exam, you can re-derive both in under ten seconds from sin²x + cos²x = 1. Treating them as independent memorized facts misses the structural unity of the three identities.

Question 4

The expression cos²(x) can always be replaced by 1 − sin²(x), regardless of the value of x.

• T. True
• F. False

Answer: True

This is a direct rearrangement of sin²x + cos²x = 1, valid for all real values of x — there are no restrictions on the domain of this identity. Two-directional fluency means recognizing that sin²x, cos²x, (1 − sin²x), and (1 − cos²x) are all interchangeable depending on which form simplifies an expression. This substitution flexibility is what makes the identities useful in calculus and in simplifying complex trigonometric expressions.

Question 5

How can the identity tan²(x) + 1 = sec²(x) be derived from sin²(x) + cos²(x) = 1? Why does knowing this derivation matter?

Model answer: Start with sin²x + cos²x = 1 and divide every term by cos²x: (sin²x/cos²x) + (cos²x/cos²x) = (1/cos²x), which gives tan²x + 1 = sec²x, since sinx/cosx = tanx and 1/cosx = secx. Knowing this derivation matters for two reasons: first, if you forget the identity you can reconstruct it instantly rather than being stuck. Second, it reveals that there are not really three separate Pythagorean identities — there is one (sin²+cos²=1) with two derived restatements, each obtained by dividing by a different squared function.

The cot²/csc² identity follows by dividing sin²x + cos²x = 1 by sin²x instead, giving 1 + cot²x = csc²x. The underlying structure is always the same: the Pythagorean theorem applied to the unit circle, rewritten in three different trigonometric 'languages.'