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.

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

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)?

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

Hello, mcleja!

You are expected to know these identities:

. .

We have: .

. .

. .

Therefore: .

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

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!

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

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 .

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

Ok i see now. Thanks everybody.