Trig Expression: (tan^3x-cot^3x)/(tan^2x+csc^2x)=tanx-cotx

[8906]

(tan^3x-cot^3x)/(tan^2x+csc^2x)=tanx-cotx

I need help verifying the above equation. I know I'm supposed to show all work I've already done, but everything I have done leads to a dead end, and I have no idea what to do. Any help would be appreciated.

 

[Mrspi]

Re: Trig Expression Help

(tan^3x-cot^3x)/(tan^2x+csc^2x)=tanx-cotx

I need help verifying the above equation. I know I'm supposed to show all work I've already done, but everything I have done leads to a dead end, and I have no idea what to do. Any help would be appreciated.

I'll help you get started.

tan[sup:32a44lbt]3[/sup:32a44lbt] x - cot[sup:32a44lbt]3[/sup:32a44lbt] x is a difference of two cubes and can be factored using this pattern:
a[sup:32a44lbt]3[/sup:32a44lbt] - b[sup:32a44lbt]3[/sup:32a44lbt] = (a - b)(a[sup:32a44lbt]2[/sup:32a44lbt] + ab + b[sup:32a44lbt]2[/sup:32a44lbt])

Notice that factoring this way produces something NICE: (tan x - cot x)

Now....if that other factor in the numerator turns out to be in the denominator also, it could be divided out.... Hmmmm....One of the Pythagorean Identities says that
csc[sup:32a44lbt]2[/sup:32a44lbt] x = 1 + cot[sup:32a44lbt]2[/sup:32a44lbt] x

Substitute 1 + cot[sup:32a44lbt]2[/sup:32a44lbt] x for the csc[sup:32a44lbt]2[/sup:32a44lbt] x in the denominator....Oh....and you might want to recall that cot x * tan x = 1.....

See if that helps.

If you get stuck, show us exactly what you've done EVEN IF IT LEADS TO A DEAD END.

 

[8906]

Re: Trig Expression Help

Thanks for that.

 

[kendallann]

you're amazing

thanks so much, you saved me! i never even thought to do a difference of cubes!!!