What Are the Sec, Csc, and Cot Pythagorean Identities?

The two reciprocal Pythagorean identities are $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\cosec^2\theta$. In IB Mathematics: Analysis and Approaches HL, cosec is standard; csc is equivalent notation for $1/\sin\theta$. They apply only where the functions are defined.

How the Identities Are Derived

Both identities follow from the fundamental Pythagorean identity:

$$\sin^2\theta+\cos^2\theta=1.$$

The strategy is to divide every term by one squared trigonometric function, then rewrite the ratios using reciprocal definitions.

Dividing by $\cos^2\theta$ gives $\frac{\sin^2\theta}{\cos^2\theta}+1=\frac{1}{\cos^2\theta}$. Since and , this becomes .

Dividing by $\sin^2\theta$ gives $1+\frac{\cos^2\theta}{\sin^2\theta}=\frac{1}{\sin^2\theta}$. Since and , this becomes .

These forms help when simplifying identities or integrating expressions containing $\sec^2\theta$. The domain restrictions remain: $\cos\theta\neq0$ for tangent and secant, and $\sin\theta\neq0$ for cotangent and cosecant.

These identities are equations, not formulas for choosing signs. Squaring removes sign information, so an identity can determine the magnitude of a reciprocal ratio but not always whether that ratio is positive or negative. Use the stated interval, quadrant, or sign of sine or cosine to finish the result.

Worked Example

If $\tan\theta=3$, then $\sec^2\theta=1+3^2=10$. Therefore, . Its sign depends on the quadrant because secant has the same sign as cosine.

For example, if $\theta$ lies in quadrant II and $\tan\theta=-2$, then secant is negative. The identity gives $\sec^2\theta=5$, and the quadrant selects $$ rather than .

Common Misconception

The statement $1+\cot^2\theta=\sec^2\theta$ is incorrect. Tangent pairs with secant, while cotangent pairs with cosecant. Another common error is dropping $\pm$ when taking a square root.

IB Exam Technique

This is AHL 3.9, so it is HL-only content and may appear on Paper 1, Paper 2, or Paper 3. In a “show” or “prove” question, begin with one side and transform it logically. State each identity used, show substitutions explicitly, and retain exact values on Paper 1. These identities also connect directly to reciprocal-function differentiation and integration in AHL calculus.