Trigonometry and method of difference

In summary, the problem is asking to prove that 1-2cosx+2cos2x-2cos3x+....+2cos8x = cos(17/2)x sec(x/2), given the equation 1+2cosx+2cos2x+2cos3x + ....+2cosnx = sin(x+1/2)x cosec(x/2). The solution suggests substituting 2x for x and using that to find the left hand side, but it is unsure if this will lead to a solution.
  • #1
gaobo9109
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Homework Statement



Given that 1+2cosx+2cos2x+2cos3x + ....+2cosnx = sin(x+1/2)x cosec(x/2)
Prove that 1-2cosx+2cos2x-2cos3x+....+2cos8x = cos(17/2)x sec(x/2)


Homework Equations





The Attempt at a Solution



1+2cosx+2cos2x+2cos3x + ....+2cos8x = sin(8+1/2)x cosec(x/2)

4cosx+4cos3x+4cos5x... = ?

I don't know how to continue. Or is there another way?
 
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  • #2
1+2cosx+2cos2x+2cos3x + ....+ 2cos8x = ... eqn(1)
Substitute 2x for x, and you can say
1 + 2cos2x + 2 cos4x + 2cos6x + 2cos8x = ... eqn(2)

If you subtract double eqn(2) from eqn(1) you'll as good as have the left hand side. I don't know whether this will get you anywhere.

Also, fix up the x that should be n in your first line.
 

Related to Trigonometry and method of difference

What is trigonometry and how is it used in mathematics?

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is used to solve problems involving right triangles, as well as to model and analyze periodic phenomena such as waves and oscillations.

What is the method of difference in trigonometry?

The method of difference in trigonometry is a technique used to find the difference between two trigonometric values. It involves using trigonometric identities and properties to simplify expressions and ultimately determine the difference between the two values.

How is the method of difference different from other trigonometric methods?

The method of difference is different from other trigonometric methods because it focuses specifically on finding the difference between two values rather than solving for a specific value or relationship. It is a useful tool for comparison and simplification in trigonometric equations.

What are some real-world applications of trigonometry and the method of difference?

Trigonometry and the method of difference are used in a variety of real-world applications, including navigation, surveying, architecture, engineering, and astronomy. They are essential tools for solving problems involving angles and distances, as well as for understanding and predicting periodic phenomena.

What are some common challenges when working with trigonometry and the method of difference?

Some common challenges when working with trigonometry and the method of difference include understanding and applying the various trigonometric identities and properties, as well as visualizing and interpreting the relationships between angles and sides in different contexts. It is important to have a solid understanding of basic trigonometric concepts and to practice regularly to overcome these challenges.

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