Question

Discuss continuity of cosine, cosecant, secant and cotangent function.

Answer 1

a)

$f\left(x\right)=\cos x$

$utC\in IR$

$LhL{lim}_{x\to c^{-}}RhL{lim}_{x\to c^{+}}f\left(x\right)$

$limf(c-h)={lim}_{4-10}\cos (c+h)$

$lim\cos (c-h)={lim}_{4-10}\cos c\cos h-\sin c\sin h$

${lim}_{4-10}\cos c\cos h+\sin c\sin h\cos c$

$=\cos c$

$\because lLh=Rhc=f\left(x\right)$

$\Rightarrow \text{cosine is continuous}$

b)

$f\left(x\right)=cosecx$

$=\frac{1}{\sin x}$

$\text{this is continuous}\forall xst\sin x\ne 0$

$\Rightarrow x\ne n{\pi }_{1}n\in \mathrm{z/}$

$\text{so, cosec}x\text{is cont all all}n\in R$

$stx\ne n{\pi }_{1}n\in \mathrm{z/}$

c)

$\sec x=\frac{1}{\cos x}$

$f\text{is cont.}\forall x\in IRst\cos x\ne 0$

$\Rightarrow x\ne (2n+1)\pi /2,n\in \mathrm{z/}$

d)

$f\left(x\right)=\cot x=\frac{\cos x}{\sin x}$

$f\text{is cont. for}aux\in IRst$

$\sin x\ne 0$

$\Rightarrow x\ne n{\pi }_{1}n\in \mathrm{z/}$

$\text{So},\cot x\text{is continuous at au not numbers except}x=n{\pi }_{1}n\in \mathrm{z/}$