Question #344549

Differentiate the trigonometric functions


1. = 2 sec x + tan x



2. y = sin x/ 1 + COS X



3. y = cos x cot x

1
Expert's answer
2022-05-25T09:34:21-0400

1.


y=(secx+tanx)=1cos2x(sinx)+1cos2xy'=(\sec x+\tan x)'=-\dfrac{1}{\cos^2 x}(-\sin x)+\dfrac{1}{\cos ^2 x}

=sinx+1cos2x=(sinx+1)sec2x=\dfrac{\sin x+1}{\cos ^2 x}=(\sin x+1)\sec ^2 x

2.


y=(sinx1+cosx)=cosx(1+cosx)+sin2x(1+cosx)2y'=(\dfrac{\sin x}{1+\cos x})'=\dfrac{\cos x(1+\cos x)+\sin ^2x}{(1+\cos x)^2}

=1+cosx(1+cosx)2=11+cosx=\dfrac{1+\cos x}{(1+\cos x)^2}=\dfrac{1}{1+\cos x}

3.


y=(cosxcotx)=sinxcotxcosxsin2xy'=(\cos x\cot x)'=-\sin x\cot x-\dfrac{\cos x}{\sin^2 x}

=cosxcotxcscx=-\cos x-\cot x\csc x


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