Trig Identities

Simplify Trig Expressions

Simplify (1 − cos²θ)/sin θ to sin θ by substituting the Pythagorean identity, verified symbolically in the Pythagorean polynomial ring.

Symbolic verification: in the ring ℚ[s,c]/(s²+c²−1), 1 − c² reduces to s², confirming the substitution is exact.

Example

Use the Pythagorean identity to simplify a trig expression. This simplification is valid only where the sine of theta is nonzero, since the canceled factor must not be zero.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

1−cos⁡2θsin⁡θ\frac{ 1 -\cos^{ 2 }\theta}{\sin\theta}

Step 2 — Use the identity

Replace 1 minus cos squared with sin squared.

1−cos⁡2θ=sin⁡2θ1 -\cos^{ 2 }\theta=\sin^{ \hl{2} }\theta

Step 3 — Substitute

Substitute sin squared over sin.

sin⁡2θsin⁡θ\frac{\sin^{ \hl{2} }\theta}{\sin\theta}

Step 4 — Cancel one sine

Cancel one sine to get sin theta.

sin⁡θ\sin\theta
simplify-trig-expression Use the Pythagorean identity sin²θ + cos²θ = 1 to replace 1 − cos²θ with sin²θ, then cancel the common sin θ factor: (1 − cos²θ)/sin θ = sin²θ/sin θ = sin θ