# Proving trigonometric functions

• MHB
• kbr1804
In summary, the conversation discusses how to prove that $6\cos(x+45)\cos(x-45)$ is equal to $3\cos(2x)$, not $3\cos{x}$. The use of the sum/difference identity for cosine is mentioned and a typo is pointed out. Additional discussion on graphing this equation on Desmos is also included.
kbr1804
How can i prove that 6cos(x+45) cos(x-45) is equal to 3cosx?

use the sum/difference identity $\cos(a \pm b) = \cos{a}\cos{b} \mp \sin{a}\sin{b}$

it should be equal to $3\cos(2x)$, not $3\cos{x}$

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yeah i think i got it lol thanks alot:)and yeah it was supposed to equal to 3cos2x that was a typo

Beer soaked ramblings follow.
kbr1804 said:
How can i prove that 6cos(x+45) cos(x-45) is equal to 3cosx?
skeeter said:
use the sum/difference identity $\cos(a \pm b) = \cos{a}\cos{b} \mp \sin{a}\sin{b}$

it should be equal to $3\cos(2x)$, not $3\cos{x}$
kbr1804 said:
yeah i think i got it lol thanks alot:)and yeah it was supposed to equal to 3cos2x that was a typo
Three graphs; shouldn't 2 of them have the same graph?
https://www.desmos.com/calculator/x3dyyfbucy

show up the same on my calculator ...

and on Desmos ...

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-180,"ymin":-10.762090536086637,"xmax":180,"ymax":10.762090536086637},"degreeMode":true,"squareAxes":false},"randomSeed":"78931067dd5aa2f77a194c669752ab59","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"6\\cos\\left(x+45\\right)\\cos\\left(x-45\\right)\\left\\{x>0\\right\\}"},{"type":"expression","id":"2","color":"#2d70b3","latex":"3\\cos\\left(2x\\right)\\left\\{x<0\\right\\}"},{"type":"expression","id":"3","color":"#388c46"}]}}[/DESMOS]

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jonah, in Desmos one should write $\pi/4$ instead of 45.

Evgeny.Makarov said:
jonah, in Desmos one should write $\pi/4$ instead of 45.

One can change to degree mode with the "wrench" button menu

Beer soaked ramblings follow.
Evgeny.Makarov said:
jonah, in Desmos one should write $\pi/4$ instead of 45.
skeeter said:
One can change to degree mode with the "wrench" button menu
Well aware of that.
Force of habit.
In the absence of the degree symbol (°), assumed that 45 was in radians.
Didn't occur to me to check it in degree mode.
Usually do it by multiplying the degree measure by $\frac{\pi}{180}$ if the expression just gives it once. Otherwise, I usually do it by skeeter's suggestion (which I'm also quite aware of).
skeeter said:
... and on Desmos ...

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-180,"ymin":-10.762090536086637,"xmax":180,"ymax":10.762090536086637},"degreeMode":true,"squareAxes":false},"randomSeed":"78931067dd5aa2f77a194c669752ab59","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"6\\cos\\left(x+45\\right)\\cos\\left(x-45\\right)\\left\\{x>0\\right\\}"},{"type":"expression","id":"2","color":"#2d70b3","latex":"3\\cos\\left(2x\\right)\\left\\{x<0\\right\\}"},{"type":"expression","id":"3","color":"#388c46"}]}}[/DESMOS]
I've often wondered what kind of platform this type of Desmos "quoting" is ever since I saw one of Klaas van Aarsen's post which used the same method. It isn't just a link to Desmos as I found out when I hit the reply tab on my phone. Is it that the TikZ thingamajig I've been seeing a lot lately on this site? I think I remember copying that stuff in another math site but was surprised that it didn't work there.

kbr1804 said:
How can i prove that 6cos(x+45) cos(x-45) is equal to 3cosx?
You CAN'T- it's not true! For example if x= 45 degrees this becomes 6 cos(90)cos(0)= 6(0)(1)= 0 but 3 cos(45)= 3sqrt(2)/2.

Country Boy said:
You CAN'T- it's not true! For example if x= 45 degrees this becomes 6 cos(90)cos(0)= 6(0)(1)= 0 but 3 cos(45)= 3sqrt(2)/2.

post #2 …

skeeter said:
use the sum/difference identity $\cos(a \pm b) = \cos{a}\cos{b} \mp \sin{a}\sin{b}$

it should be equal to $3\cos(2x)$, not $3\cos{x}$

## What are trigonometric functions?

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent.

## Why is it important to prove trigonometric functions?

Proving trigonometric functions allows us to understand the relationships between angles and sides in a triangle and use them to solve complex geometric problems. It also helps to verify the accuracy of mathematical calculations involving trigonometric functions.

## What are the methods used to prove trigonometric functions?

The most common methods used to prove trigonometric functions are the unit circle method, the right triangle method, and the algebraic method. These methods involve using geometric principles, trigonometric identities, and algebraic manipulations to prove the desired function.

## What are some common trigonometric identities used in proving functions?

Some common trigonometric identities used in proving functions include the Pythagorean identities, the double angle identities, and the sum and difference identities. These identities help to simplify complex trigonometric expressions and establish relationships between different functions.

## Are there any tips for proving trigonometric functions?

Yes, some tips for proving trigonometric functions include understanding the properties of triangles, memorizing key identities, and practicing with different types of problems. It is also important to carefully follow the steps of the chosen method and double-check the solution for accuracy.

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