Thread: Trigonometric rearrangement of tan(3x)cot(7x) to find limit

I have tan(3x)cot(7x) and i need to get it in the form (sin(ax)/ax)/(sin(bx)/bx) for some a and b. I have use the tan(x)=sin(x)/cos(x) and cot(x)=cos(x)/sin(x) to get (sin(3x)/cos(3x))(cos(7x)/sin(7x)) and that is about as far as i got.
Could anybody help me to solve this rearrangement. Then I need to find the limit as x goes to zero. But i have to show it in the form stated above.

Thanks.

Could one use L'hopitals rule since (sin(3x)/cos(3x))(cos(7x)/sin(7x)) will equal 0/0. I tried this and got (sin(4x)(-7sin(7x))+4cos(7x)cos(4x))/(cos(4x)7cos(7x)+sin(7x)(-4sin(4x)) and i yo substitute in 0 you get 4/7 which i have confirmed is the right limit. But how do you go from my result to the form (sin(ax)/ax)/(sin(bx)/bx)?

Hello, mcleja!

You are expected to know these identities:

. .

We have: .

. .

. .


Therefore: .

ok i got that as the limit correctly but i could not get the expression tan(3x)cot(7x) in the form (sin(ax)/ax)/(sin(bx)/bx) which i need to show as an intermediate step for my assignment. Could you help me do that? Thank you for the reply!

You're not going to be able to get rid of the cosine terms, I expect that you are asked to have that form in your expression AMONG OTHER THINGS.

Anyway, notice that is equivalent to .

Ok i see now. Thanks everybody.