The quotient identities express tangent and cotangent in terms of sine and cosine (tan theta equals sin over cos). The reciprocal identities pair cosecant, secant, and cotangent with sine, cosine, and tangent as flipped ratios. Together they let you rewrite any trig expression using only sine and cosine.
The quotient identities
The quotient identities rewrite tangent and cotangent in terms of sine and cosine. Since tan θ is defined as the ratio of the opposite side to the adjacent side, and sin θ and cos θ carry that same ratio information, tangent turns out to equal sin θ divided by cos θ. Cotangent is the reciprocal ratio, so it equals cos θ divided by sin θ.
The quotient identities rewrite tan and cot using sin and cos; the reciprocal identities pair csc, sec, and cot with sin, cos, and tan.
The reciprocal identities
The reciprocal identities connect the three main trig ratios to their flipped-over partners: cosecant is 1 over sine, secant is 1 over cosine, and cotangent is 1 over tangent. Because sine and cosine can never be exactly 0 or undefined in the same place, these reciprocal ratios are undefined only where their partner ratio is 0.
Why these identities matter
Quotient and reciprocal identities let you rewrite any trig expression using only sine and cosine, which is often the easiest way to simplify an expression or verify that two sides of an equation are equal. They also connect directly to the Pythagorean identities and the sum and difference identities, since many proofs substitute a quotient or reciprocal form partway through.
Worked example
Simplify tan θ · cos θ. Substitute the quotient identity: tan θ · cos θ = (sin θ / cos θ) · cos θ = sin θ. The cos θ terms cancel, leaving a simpler expression.