Reduce Trigonometric Series: 2 Problems Solved

In summary: I am stuck at this part!tan (x/2)=2 tan (x/4) / 1- tan^2 (x/4)Can this ever help? I really have been spending a wretched time to solve thatIn summary, the homeworks statement is that there are two problems with sin and tan. Sin and tan can be simplified to y=2n+1 and tan(x/2)=2 tan (x/4) / 1- tan^2 (x/4), respectively. However, in the first problem, y=2^{(n+1)} doesn't seem to make sense and in the second problem, the denominator for sec2x is difficult
  • #1
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Homework Statement



Reduce the following series to the most simplified form:

1. 2 sin[tex]^{2}[/tex]([tex]\theta[/tex]/2)*(1+cos[tex]\theta[/tex])*(1+cos[tex]^{2}[/tex][tex]\theta[/tex])(1+cos[tex]^{4}[/tex][tex]\theta[/tex])...

2. tan ([tex]\theta[/tex]/2)*(1+sec [tex]\theta[/tex])(1+2[tex]\theta[/tex])(1+sec 4[tex]\theta[/tex]).......

Homework Equations


For problem 1:
2 sin[tex]^{2}[/tex]([tex]\theta[/tex]/2)=1-cos[tex]\theta[/tex])

The Attempt at a Solution



For problem 1:
the result I got:
y=2[tex]^{(n+1)}[/tex]

Result=
1-cos[tex]^{y}[/tex][tex]\theta[/tex]




is it correct?
For problem 2:
If I try to expand
sec 2 [tex]\theta[/tex]
I get huge denominators which are difficult to solve. Similarly with tan ([tex]\theta[/tex]/2),I again get a denominator with no pattern following any rule.




Please help me for the second one. and tell me if I am correct in solving the first problem.
regards,
Ritwik
 
Last edited:
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  • #2
So I have finally got that latex code right. I have been trying the second problem for hours. Please help me.
Thanks in advance.
Regards,
Ritwik
 
  • #3
ritwik06 said:

Homework Statement



Reduce the following series to the most simplified form:

1. 2 sin[tex]^{2}[/tex]([tex]\theta[/tex]/2)*(1+cos[tex]\theta[/tex])*(1+cos[tex]^{2}[/tex][tex]\theta[/tex])(1+cos[tex]^{4}[/tex][tex]\theta[/tex])...

2. tan ([tex]\theta[/tex]/2)*(1+sec [tex]\theta[/tex])(1+2[tex]\theta[/tex])(1+sec 4[tex]\theta[/tex]).......
I presume you mean
tan ([tex]\theta[/tex]/2)*(1+sec [tex]\theta[/tex])(1+sec(2[tex]\theta[/tex]))(1+sec 4[tex]\theta[/tex])

Homework Equations


For problem 1:
2 sin[tex]^{2}[/tex]([tex]\theta[/tex]/2)=1-cos[tex]\theta[/tex])

The Attempt at a Solution



For problem 1:
the result I got:
y=2[tex]^{n+1}[/tex]
Result=
1-cos[tex]^{y}[/tex][tex]\theta[/tex]




is it correct?
There is no "n" in the problem so I have no idea what "y= 2n+1" means!

For problem 2:
If I try to expand
sec 2 [tex]\theta[/tex]
I get huge denominators which are difficult to solve. Similarly with tan ([tex]\theta[/tex]/2),I again get a denominator with no pattern following any rule.




Please help me for the second one. and tell me if I am correct in solving the first problem.
regards,
Ritwik
Since you say "huge denominators" I presume you switched to sine and cosine. That shouldn't be necessary. Use the corresponding identities as in (1) for tangent and secant.
 
  • #4
HallsofIvy said:
I presume you mean
tan ([tex]\theta[/tex]/2)*(1+sec [tex]\theta[/tex])(1+sec(2[tex]\theta[/tex]))(1+sec 4[tex]\theta[/tex])


There is no "n" in the problem so I have no idea what "y= 2n+1" means!
'n' stands for the number of terms
Since you say "huge denominators" I presume you switched to sine and cosine. That shouldn't be necessary. Use the corresponding identities as in (1) for tangent and secant.
Yes I switched to sine and cosine. But I wonder which idntitis to use. I have been working over thse problms for 6 hours now. Could u please help?
Thanks
 
  • #5
ritwik06 said:
'n' stands for the number of terms

Yes I switched to sine and cosine. But I wonder which idntitis to use. I have been working over thse problms for 6 hours now. Could u please help?
Thanks

can this ever help 1+tan^2=sec^2
this is the only identity i know between sec and tan. Please help me someone
 
  • #6
I really have been spending a wretched time to solve that
tan (x/2)*(1+sec x)(1+sec 2x)(1+sec 4x).......
How can I apply any identity here??
tan (x/2)=2 tan (x/4) / 1- tan^2 (x/4)
Please help!
 
Last edited:

1. What is a trigonometric series?

A trigonometric series is a mathematical series that involves sine and cosine functions. It can be written in the form of a sum of terms involving these trigonometric functions and their coefficients.

2. Why would someone want to reduce a trigonometric series?

Reducing a trigonometric series can make it easier to work with and solve mathematical problems. It can also help simplify complex expressions and make them more manageable.

3. How do you reduce a trigonometric series?

To reduce a trigonometric series, you can use various mathematical techniques such as trigonometric identities, substitution, and manipulation of series. The specific method used will depend on the specific series and problem at hand.

4. What are some common problems that can be solved by reducing a trigonometric series?

Some common problems that can be solved by reducing a trigonometric series include finding the sum of a series, evaluating integrals, solving differential equations, and solving problems in physics and engineering that involve periodic functions.

5. Are there any limitations to reducing a trigonometric series?

Yes, there are certain limitations to reducing a trigonometric series. For example, some series may not have a closed form solution or may require advanced mathematical techniques to reduce. Additionally, reducing a series may not always lead to an exact solution and may result in approximations.

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