# What is the derivative of #y=(1-sec(x))/tan(x)#?

This problem can be solved by two methods, as mentioned below :

Explanation (I), Simplifying the expression

Explanation (II)

This can also be solved using Quotient Rule

Which is ,

In same way, for the problem,

If we club first and last term,

Simplifying further, we get

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To find the derivative of ( y = \frac{1 - \sec(x)}{\tan(x)} ), you can apply the quotient rule. The quotient rule states that if you have a function in the form ( \frac{u}{v} ), where ( u ) and ( v ) are functions of ( x ), then the derivative is given by:

[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ]

Let ( u = 1 - \sec(x) ) and ( v = \tan(x) ). Then:

[ \frac{du}{dx} = 0 - (-\sec(x)\tan(x)) = \sec(x)\tan(x) ] [ \frac{dv}{dx} = \sec^2(x) ]

Now, apply the quotient rule:

[ \frac{d}{dx}\left(\frac{1 - \sec(x)}{\tan(x)}\right) = \frac{\tan(x) \cdot \sec(x)\tan(x) - (1 - \sec(x)) \cdot \sec^2(x)}{\tan^2(x)} ]

Simplify:

[ \frac{\sec(x)\tan^2(x) - \sec(x)\tan(x) - \sec^3(x) + \sec^2(x)}{\tan^2(x)} ]

[ \frac{\sec(x)(\tan^2(x) - \tan(x)) - \sec^2(x)(\tan^2(x) - 1)}{\tan^2(x)} ]

[ \frac{\sec(x)(\tan^2(x) - \tan(x)) - \sec^2(x)\tan^2(x) + \sec^2(x)}{\tan^2(x)} ]

[ \frac{\sec(x)(\tan^2(x) - \tan(x)) + \sec^2(x)(1 - \tan^2(x))}{\tan^2(x)} ]

[ \frac{\sec(x)(\tan(x) - \tan^2(x)) + \sec^2(x)(1 - \tan^2(x))}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x)(1 - \tan(x)) + \sec^2(x)(1 - \tan^2(x))}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x)(1 - \tan(x)) + \sec^2(x)(\sec^2(x) - 1)}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x)(1 - \tan(x)) + \sec^4(x) - \sec^2(x)}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x) - \sec^2(x)\tan(x) + \sec^4(x) - \sec^2(x)}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x) - \sec^2(x)(\tan(x) - 1) - \sec^2(x)}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x) - \sec^2(x)\tan(x) + \sec^2(x) - \sec^2(x)}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x) - \sec^2(x)\tan(x)}{\tan^2(x)} ]

[ \frac{\sec(x)\tan(x)(1 - \sec(x))}{\tan^2(x)} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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