Reciprocal trigonometric functions

Objective. Use secant, cosecant and cotangent identities.

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Learn: the key idea

Reciprocal trigonometric functions secant, cosecant and cotangent are defined as sec x=1/cos x, cosec x=1/sin x and cot x=1/tan x, each undefined where the corresponding sine, cosine or tangent is zero. Example: the identity 1+tan²x=sec²x follows from dividing sin²x+cos²x=1 by cos²x, and similarly 1+cot²x=cosec²x follows by dividing by sin²x. These identities are essential for simplifying expressions and solving equations involving reciprocal functions.

Worked example

Solve sec²x−3tan x=1 for 0≤x≤360°, using reciprocal trigonometric identities.

1 $Replace sec²x with 1+tan²x to get 1+tan²x-3tan x=1$
2 $Simplify to tan²x-3tan x=0$
3 $Factorise: tan x(tan x-3)=0, so tan x=0 or tan x=3$
4 $Solve each: x=0°,180°,360° from tan x=0, and x=71.6°,251.6° from tan x=3$

Answer: x=0°, 71.6°, 180°, 251.6°, 360°

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Common mistake. Students write sec x=1/sin x, confusing it with cosec x; remember sec pairs with cos and cosec pairs with sin, matching the first letters after the 'co-' prefix carefully.

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