Find the equivalent expression

Find the equivalent expression. csc x csc x cot x Csc x(csc * cot x) sec x(sec * tan x) CSC_ CSC * cot x) d. sec x(sec * tan x) sec x(sec * cot.)

The Correct Answer and Explanation is:

To clarify your question, you’re asking to find the equivalent expression for a given trigonometric expression involving cosecant (csc), cotangent (cot), and secant (sec) functions. Here’s how we can break it down and simplify the expression:

Expression:

csc⁡(x)⋅csc⁡(x)⋅cot⁡(x)\csc(x) \cdot \csc(x) \cdot \cot(x)csc(x)⋅csc(x)⋅cot(x)

This simplifies as: csc⁡2(x)⋅cot⁡(x)\csc^2(x) \cdot \cot(x)csc2(x)⋅cot(x)

Now, let’s rewrite it in a more manageable form for simplification and identify any possible trigonometric identities.

We can use some standard trigonometric identities here:

• csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}csc(x)=sin(x)1​
• cot⁡(x)=cos⁡(x)sin⁡(x)\cot(x) = \frac{\cos(x)}{\sin(x)}cot(x)=sin(x)cos(x)​

Thus, the expression becomes: csc⁡2(x)⋅cot⁡(x)=(1sin⁡2(x))⋅(cos⁡(x)sin⁡(x))\csc^2(x) \cdot \cot(x) = \left( \frac{1}{\sin^2(x)} \right) \cdot \left( \frac{\cos(x)}{\sin(x)} \right)csc2(x)⋅cot(x)=(sin2(x)1​)⋅(sin(x)cos(x)​)

Simplifying further: =cos⁡(x)sin⁡3(x)= \frac{\cos(x)}{\sin^3(x)}=sin3(x)cos(x)​

Next Expression: sec⁡(x)⋅sec⁡(x)⋅tan⁡(x)\sec(x) \cdot \sec(x) \cdot \tan(x)sec(x)⋅sec(x)⋅tan(x)

Now, for the second part of your question: sec⁡2(x)⋅tan⁡(x)\sec^2(x) \cdot \tan(x)sec2(x)⋅tan(x)

Using the identity:

• sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}sec(x)=cos(x)1​
• tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)​

This expression becomes: sec⁡2(x)⋅tan⁡(x)=(1cos⁡2(x))⋅(sin⁡(x)cos⁡(x))\sec^2(x) \cdot \tan(x) = \left( \frac{1}{\cos^2(x)} \right) \cdot \left( \frac{\sin(x)}{\cos(x)} \right)sec2(x)⋅tan(x)=(cos2(x)1​)⋅(cos(x)sin(x)​)

Simplifying: =sin⁡(x)cos⁡3(x)= \frac{\sin(x)}{\cos^3(x)}=cos3(x)sin(x)​

Summary of Equivalent Expressions:

1. For csc⁡2(x)⋅cot⁡(x)\csc^2(x) \cdot \cot(x)csc2(x)⋅cot(x), the equivalent expression is: cos⁡(x)sin⁡3(x)\frac{\cos(x)}{\sin^3(x)}sin3(x)cos(x)​
2. For sec⁡2(x)⋅tan⁡(x)\sec^2(x) \cdot \tan(x)sec2(x)⋅tan(x), the equivalent expression is: sin⁡(x)cos⁡3(x)\frac{\sin(x)}{\cos^3(x)}cos3(x)sin(x)​

These two expressions are now simplified forms of the original ones you provided.