13. rate of change in calculus, it can be shown that the rate of change of the function $f(x)=csc x-cot x$…

13. rate of change in calculus, it can be shown that the rate of change of the function $f(x)=csc x-cot x$ is given by the expression: $\frac{1 - cos x}{sin^{2}x}$. show that this expression for the rate of change can be written as $csc^{2}x-csc xcot x$.

13. rate of change in calculus, it can be shown that the rate of change of the function $f(x)=csc x-cot x$ is given by the expression: $\frac{1 - cos x}{sin^{2}x}$. show that this expression for the rate of change can be written as $csc^{2}x-csc xcot x$.

Answer

Explanation:

Step1: Recall trigonometric identities

We know that $\csc x=\frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$.

Step2: Rewrite the given expression

We start with $\frac{1 - \cos x}{\sin^{2}x}$. Split it into two - fractions: $\frac{1}{\sin^{2}x}-\frac{\cos x}{\sin^{2}x}$.

Step3: Simplify using identities

Since $\frac{1}{\sin^{2}x}=\csc^{2}x$ and $\frac{\cos x}{\sin^{2}x}=\frac{\cos x}{\sin x}\cdot\frac{1}{\sin x}=\csc x\cot x$, the expression $\frac{1}{\sin^{2}x}-\frac{\cos x}{\sin^{2}x}$ can be written as $\csc^{2}x-\csc x\cot x$.

Answer:

We have shown that $\frac{1 - \cos x}{\sin^{2}x}=\csc^{2}x-\csc x\cot x$ by using the trigonometric identities $\csc x=\frac{1}{\sin x}$ and $\cot x=\frac{\cos x}{\sin x}$ and splitting the given fraction into two parts and simplifying.