What is the proof for sin(45+x).sin(45-x) = \frac{1}{2}cos2x?

In summary, using double angle formulae and special angles, it can be shown that sin(45+x).sin(45-x) = \frac{1}{2}cos2x.
  • #1
DERRAN
34
0

Homework Statement


Prove that sin(45+x).sin(45-x) = [tex]\frac{1}{2}[/tex]cos2x


Homework Equations


double angle formulae
reduction formulae
special angles
identities


The Attempt at a Solution


(sin45.cosx+cos45.sinx)(sin45.cosx-cos45.sinx)
=(sin45.cosx-cos45.sinx)2
=([tex]\frac{1}{\sqrt{2}}[/tex]cosx-[tex]\frac{1}{\sqrt{2}}[/tex]sinx)2

=[tex]\frac{1}{2}[/tex]cos2x-sinx.cosx+[tex]\frac{1}{2}[/tex]sinx2

=[tex]\frac{1}{2}[/tex]-sinx.cosx
 
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  • #2
How did you get from:

(sin45.cosx+cos45.sinx)(sin45.cosx-cos45.sinx)

To:

(sin45.cosx-cos45.sinx)^2

?
 
  • #3
danago said:
How did you get from:

(sin45.cosx+cos45.sinx)(sin45.cosx-cos45.sinx)

To:

(sin45.cosx-cos45.sinx)^2

?
I really don't know. I must of been smoking something(hehehe):smile:
But thanks any way for pointing out my error I got it now.

it should be

(sin45.cosx+cos45.sinx)(sin45.cosx-cos45.sinx)
=sin245.cos2x-cos245.sin2x
=[tex]\frac{1}{2}[/tex]cos2x-[tex]\frac{1}{2}[/tex]sin2x
=[tex]\frac{1}{2}[/tex]cos2x
 
Last edited:
  • #4
Start from here:

http://http://i623.photobucket.com/albums/tt316/Saxifrage_Russell/PhysicsForumcomMarch21st.png"

PhysicsForumcomMarch21st.png
 
Last edited by a moderator:

Related to What is the proof for sin(45+x).sin(45-x) = \frac{1}{2}cos2x?

What are the basic trigonometric ratios?

The basic trigonometric ratios are the sine, cosine, and tangent. These ratios are defined as the ratio of the sides of a right triangle, with the hypotenuse being the longest side, the opposite side being the side opposite the angle of interest, and the adjacent side being the side adjacent to the angle of interest.

How do you find the value of a trigonometric ratio?

To find the value of a trigonometric ratio, you must know the length of two sides of a right triangle. The ratio can then be calculated by dividing the length of the opposite or adjacent side by the length of the hypotenuse.

What are double angle identities?

Double angle identities are trigonometric identities that involve doubling the measure of an angle. These identities are useful for simplifying trigonometric expressions and solving equations involving trigonometric functions.

How do you use double angle identities?

To use double angle identities, you must first identify the angle that needs to be doubled. Then, use the appropriate identity to simplify the expression or equation. It is important to remember to substitute the original angle back into the simplified expression to get the final answer.

What is the relationship between trigonometric ratios and double angles?

The relationship between trigonometric ratios and double angles is that double angle identities can be used to express trigonometric ratios in terms of double angles. This can be helpful in solving equations involving trigonometric ratios or finding the value of a trigonometric ratio for a given double angle.

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