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At which angles are tangent and secant undefined?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In trigonometry, both the tangent and secant functions are undefined at angles where the cosine of the angle is equal to zero.

Because the tangent function is defined as $\tan A = \frac{\sin A}{\cos A}$ and the secant function is the reciprocal of the cosine ( $\sec A = \frac{1}{\cos A}$ ), division by zero occurs whenever $\cos A = 0$ . This happens at angles defined by the expression:

$A = \frac{π}{2} + n π$

where $n$ is any integer. At these values, the functions exhibit vertical asymptotes and diverge toward infinity.12