How do you simplify csc x sin x/ cot x

How do you simplify csc x sin x/ cot x

The Correct Answer and Explanation is:

To simplify the expression csc⁡xsin⁡xcot⁡x\frac{\csc x \sin x}{\cot x}cotxcscxsinx​, we can follow these steps:

Step 1: Recall the Trigonometric Identities

First, we should recall the basic trigonometric identities:

• csc⁡x=1sin⁡x\csc x = \frac{1}{\sin x}cscx=sinx1​
• cot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}cotx=sinxcosx​

Now, we can rewrite the expression by substituting these identities: csc⁡xsin⁡xcot⁡x=(1sin⁡x)sin⁡xcos⁡xsin⁡x\frac{\csc x \sin x}{\cot x} = \frac{\left(\frac{1}{\sin x}\right) \sin x}{\frac{\cos x}{\sin x}}cotxcscxsinx​=sinxcosx​(sinx1​)sinx​

Step 2: Simplify the Numerator

In the numerator, we have 1sin⁡x×sin⁡x\frac{1}{\sin x} \times \sin xsinx1​×sinx. The sin⁡x\sin xsinx terms cancel each other out, leaving us with: 11=1\frac{1}{1} = 111​=1

So now the expression becomes: 1cos⁡xsin⁡x\frac{1}{\frac{\cos x}{\sin x}}sinxcosx​1​

Step 3: Simplify the Denominator

Now, simplify the fraction in the denominator. Dividing by cos⁡xsin⁡x\frac{\cos x}{\sin x}sinxcosx​ is equivalent to multiplying by its reciprocal: 1cos⁡xsin⁡x=sin⁡xcos⁡x\frac{1}{\frac{\cos x}{\sin x}} = \frac{\sin x}{\cos x}sinxcosx​1​=cosxsinx​

Step 4: Final Simplification

Now, sin⁡xcos⁡x\frac{\sin x}{\cos x}cosxsinx​ is just the definition of tan⁡x\tan xtanx: sin⁡xcos⁡x=tan⁡x\frac{\sin x}{\cos x} = \tan xcosxsinx​=tanx

So, the simplified expression is: tan⁡x\tan xtanx

Conclusion

Therefore, the expression csc⁡xsin⁡xcot⁡x\frac{\csc x \sin x}{\cot x}cotxcscxsinx​ simplifies to tan⁡x\tan xtanx. This process involved using basic trigonometric identities to eliminate terms and simplify the fraction step by step.