Verifying Trig Identities

[iuhn]

Hello! I'm having a really tough time wrapping my head around this.

"Verify that each of the following is an identity."
"1 - cot⁴x = 2csc²x - csc⁴x"

Help is much appreciated! :]

 

[Deleted member 4993]

Hello! I'm having a really tough time wrapping my head around this.

"Verify that each of the following is an identity."
"1 - cot⁴x = 2csc²x - csc⁴x"

Help is much appreciated! :]

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Please share your work/thoughts about this assignment

Use difference of squares to factorize LHS and then use the following identity:

1 + cot²(x) = csc²(x)

Continue......

 

[iuhn]

Ah well, I originally tried to change the RHS into the left since the left looked simpler, but I didn't realize I could factor using the difference of squares. This is what I did next; I didn't get very far (also sorry for the formatting):

(1 - cot^2[x]) * (1 + cot^2[x])

csc^2(x) * (1 - cot^2[x])

csc^2(x) - csc^2(x) * cot^2(x)

And then I didn't know where to go from there. I wondered if I had to turn 1 into sin^2(x) + cos^2(x)? Or if there was another trigonometric identity I missed?

 

[Deleted member 4993]

Ah well, I originally tried to change the RHS into the left since the left looked simpler, but I didn't realize I could factor using the difference of squares. This is what I did next; I didn't get very far (also sorry for the formatting):

(1 - cot^2[x]) * (1 + cot^2[x])

csc^2(x) * (1 - cot^2[x])

csc^2(x) - csc^2(x) * cot^2(x)

And then I didn't know where to go from there. I wondered if I had to turn 1 into sin^2(x) + cos^2(x)? Or if there was another trigonometric identity I missed?

csc^2(x) * (1 - cot^2[x])

csc^2(x) * {1 - (csc^2[x] - 1)}

continue.....

 

[iuhn]

csc^2(x) * {1 - (csc^2[x] - 1))}

csc^2(x) - csc^2(x) * [csc^2(x) - 1]

csc^2(x) - [csc^4(x) - csc^2(x)]

csc^2(x) - csc^4(x) + csc^2(x)

2csc^2(x) - csc^4(x)

Thank you! So would you say the general method is to turn the side you're changing into terms of the other side? Or looking for common identities to replace certain terms on the side you're changing with?

 

[Deleted member 4993]

Thank you! So would you say the general method is to turn the side you're changing into terms of the other side? Or looking for common identities to replace certain terms on the side you're changing with?

I CANNOT say I have observed any general method for the "most efficient" path in these types of problems. I generally stare at the problem a bit - while going through some relevant trig identities - in my mind.

 

[iuhn]

I CANNOT say I have observed any general method for the "most efficient" path in these types of problems. I generally stare at the problem a bit - while going through some relevant trig identities.

Ah okays. Well thank you again Subhotosh Khan, and have a nice rest of your day! ?

 

[Steven G]

So would you say the general method is to turn the side you're changing into terms of the other side? Or looking for common identities to replace certain terms on the side you're changing with?

So would you say the general method is to turn the side you're changing into terms of the other side? I am not sure why you would ask this. Of course you need to change one side to have the same terms as the other side. How else would you get the other side otherwise.
Subotosh is correct when he said that there are no set rules in proving these identities. You just need experience, so do lots of them.

 

[iuhn]

So would you say the general method is to turn the side you're changing into terms of the other side? I am not sure why you would ask this. Of course you need to change one side to have the same terms as the other side. How else would you get the other side otherwise.
Subotosh is correct when he said that there are no set rules in proving these identities. You just need experience, so do lots of them.

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