# Discuss the continuity of the cosine, cosecant, secant, and cotangent functions

**Solution:**

It is known that if g and h are two continuous functions, then

(i) h(x) g(x), g(x) ≠ 0 is continuous.

(ii) 1g(x),g(x) ≠ 0 is continuous.

(iii) 1h(x), h(x) ≠ 0 is continuous.

Let g(x) = and h(x) = cos x are continuous functions.

It is evident that g(x) = sin x is defined for every real number.

Let c* *be a real number.

Put x = c + h

If x→c, then h→0

g(c) = sin c

lim_{x→c} g(x) = lim_{x→c} sinx = lim_{h→0} sin(c + h)

= lim_{h→0} [sin c cos h + cos c sin h]

= lim_{h→0} (sin c cos h) + limh→0 (cos c sin h)

= sin c cos 0 + cos c sin 0

= sin c (1) + cos c (0)

= sin c

⇒ lim_{x→c} g(x) = g(c)

Therefore, g(x) = sinx is a continuous function.

Let h(x) = cosx

It is evident that h(x) = cos x is defined for every real number.

Let *c *be a real number. Put x = c + h

If x→c, then h→0

h(c) = cos c

lim_{x→c} h(x) = lim_{x→c} cos x

= lim_{h→0} cos(c + h) = lim_{h→0} [cos c cos h − sinc sin h]

= lim_{h→0} (cosc cosh) − lim_{h→0} (sin c sin h)

= cos c cos 0 − sinc sin 0

= cos c(1) − sin c(0)

= cos c

⇒ lim_{x→c} h(x) = h(c)

Therefore, h(x) = cos x is a continuous function.

Therefore, it can be concluded that,

cosec x = 1/sin x, sin x ≠ 0 is continuous.

⇒ cosec x, x ≠ nπ (n ∈ Z) is continuous.

Therefore,

cosecant is continuous except at x = nπ (n ∈ Z)

sec x = 1/cos x, cos x ≠ 0 is continuous.

⇒ sec x, x ≠ (2n + 1) π / 2 (n ∈ Z) is continuous.

Therefore, secant is continuous except at x = (2n + 1) π / 2 (n∈Z)

cot x = cos x / sin x, sin x ≠ 0 is continuous.

⇒ cot x, x ≠ nπ (n ∈ Z) is continuous.

Therefore, cotangent is continuous except at x = nπ (n ∈ Z)

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 22

## Discuss the continuity of the cosine, cosecant, secant, and cotangent functions

**Summary:**

Hence we can conclude that sin x is a continuous function. cos x is a continuous function. cosecant is continuous except at x = nπ (n ∈ Z). secant is continuous except at x = (2n + 1) π / 2 (n∈Z) .cotangent is continuous except at x = nπ (n ∈ Z)

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