Pythagorean and Reciprocal Identities

Where Do the Identities Come From?

All trigonometric identities in this lesson flow from one source: the unit circle. A point on the unit circle has coordinates (cosθ, sinθ). Since it lies on the circle x² + y² = 1, we immediately get sin²θ + cos²θ = 1. This is not a formula to memorise in isolation — it is a direct statement of Pythagoras’s theorem applied to the unit circle, and everything else follows from it by algebraic manipulation.

The Reciprocal Functions

The reciprocal functions secant, cosecant, and cotangent are defined purely for convenience — they let us write expressions more compactly and avoid repeated fractions. It is crucial to know which function is the reciprocal of which:

secθ = 1/cosθ     (secant is the reciprocal of cosine)

cosecθ = 1/sinθ     (cosecant is the reciprocal of sine)

cotθ = cosθ/sinθ = 1/tanθ     (cotangent is the reciprocal of tangent)

A common memory error is confusing sec and cosec. Remember: sec goes with cos (both contain the letter ‘c’ immediately after sec/cos).

Deriving the Other Pythagorean Identities

Starting from sin²θ + cos²θ = 1, we can divide through by cos²θ (wherever cosθ ≠ 0) to obtain:

sin²θ/cos²θ + 1 = 1/cos²θ

tan²θ + 1 = sec²θ

Similarly, dividing by sin²θ (wherever sinθ ≠ 0):

1 + cot²θ = cosec²θ

There are therefore three Pythagorean identities, all equivalent to each other. You should be able to move between them fluently.

Strategy for Simplifying Expressions

When simplifying or proving trig identities, build a mental toolkit of substitutions. Common strategies include:

1. Replace 1 − cos²θ with sin²θ (or vice versa).

2. Replace 1 + tan²θ with sec²θ.

3. Convert all functions to sin and cos — this is a reliable fallback strategy that always works, even if it is not always the most elegant.

4. Factorise — for example, sin⁴θ − cos⁴θ = (sin²θ + cos²θ)(sin²θ − cos²θ) = 1 × (sin²θ − cos²θ).

Proving Identities: The Golden Rule

The golden rule of proving identities is: work on each side independently. You cannot add, subtract, multiply, or divide both sides by the same thing (that would assume the identity is true before you prove it). Typically, start with the more complicated side and transform it, step by step, until it matches the simpler side. Each step must be a valid algebraic manipulation or a known identity substitution.

Common approaches: convert everything to sin and cos; look for Pythagorean identity patterns; factorise differences of squares; use algebra (multiply top and bottom by a conjugate expression).

Expressing One Ratio in Terms of Another

Given partial information about a trig ratio (for example, sinθ = 3/5 in the first quadrant), we can find all other ratios using the Pythagorean identities and CAST rule. For instance, sin²θ + cos²θ = 1 gives cosθ = ±√(1 − sin²θ), and the quadrant tells us the sign.