If f(x)=sec2 x+cosec 2 x, then f(x) is discontinuous at all...

If $f(x)=\sec 2 x+\operatorname{cosec} 2 x$, then $f(x)$ is discontinuous at all points in
(a) $\{n \pi, n \in N\}$
(b) $\left\{(2 n \pm 1) \frac{\pi}{4}, n \in I\right\}$
(c) $\left\{\frac{n \pi}{4}, n \in I\right\}$
(d) $\left\{(2 n \pm 1) \frac{\pi}{8}, n \in I\right\}$

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions, such as secant and cosecant, are defined as the inverses of cosine and sine respectively. Because these functions involve taking 1 divided by another trigonometric function, they become undefined whenever the base function (sine or cosine) is zero, leading to discontinuities in their graphs.

Points of Discontinuity in Functions

A function is discontinuous at a point where it is not defined, often due to a denominator being zero. For functions involving reciprocals, determining the zeros of the underlying trigonometric functions is crucial, as these are the points where the original function fails to be defined.

Periodicity of Trigonometric Functions

Trigonometric functions have periodic behavior, meaning their values repeat over regular intervals. This periodic nature means that the zeros of sine and cosine, and consequently the discontinuities of their reciprocal functions, will also occur periodically across the real number line.

Domain Restrictions in Trigonometric Contexts

When dealing with trigonometric functions, particularly those involving reciprocals, it is essential to identify and exclude the points where the function's denominator is zero. This process of determining the valid input values is a key step in understanding the overall behavior and continuity of the function.

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