Question

using the rule for differentiation for quotient of two functions prove that d/dx (sinx/cosx)=sec^2x​

Answer 1

Answer:

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Answer 2

Given that,

The two function equation is

$\dfrac{d}{dx}(\dfrac{\sin x}{\cos x})=\sec^2x$

We need to prove that $\dfrac{d}{dx}\dfrac{\sin x}{\cos x}=\sec^2x$

Using L.H.S from given equation

$=\dfrac{d}{dx}(\dfrac{\sin x}{\cos x})$

On differentiating w.r.to x in R.H.S

$=\dfrac{cos x\dfrac{d}{dx}(\sin x)-\sin\dfrac{d}{dx}(\cos x)}{cos^2x}$

$=\dfrac{\cos x\times\cos x-\sin x\times(-\sin x)}{\cos^2 x}$

$=\dfrac{\cos^2x+\sin^2x}{\cos^2x}$

We know that,

$\cos^2x+\sin^2x=1$

Put the value of $\cos^2x+\sin^2x$

$=\dfrac{1}{\cos^2x}$

$=\sec^2x$

$=R.H.S$

Hence, L.H.S=R.H.S

This is required solution.