# Trig Identity

by cscott
Tags: identity, trig
 P: 786 Can someone please help me establish this identity? $$\cos \theta (\tan \theta + \cot \theta) = \csc \theta$$
 P: 91 So, are you proving this identity? Express your tangent and cotangent in terms of sine and cosine. Get their LCD... and your numerator becomes a well-known trigonometric identity.. Can you continue from here? :D
P: 786
 Quote by irony of truth So, are you proving this identity? Express your tangent and cotangent in terms of sine and cosine. Get their LCD... and your numerator becomes a well-known trigonometric identity.. Can you continue from here? :D
The easy ones always get me :\

Thanks!

 P: 786 Trig Identity I can't get this one either: $$\frac{1 + \tan \theta}{1 - \tan \theta} = \frac{\cot \theta + 1}{\cot \theta - 1}$$ I'm so bad at proofs
 HW Helper P: 660 For this one, you can either choose to replace tan x by 1/cot x or replace cot x by 1/tan x. Choose either and do some algebriac manipulations while leaving the other side alone.
 HW Helper P: 1,021 Or, if that doesn't work for you, substitute tan by sin/cos and cot by cos/sin, then simplify the expressions Try, if you get stuck, show us!
 P: 786 I end up with $$\frac{\cos^2 \theta + \sin \theta \cos \theta}{\cos^2 \theta - \sin \theta \cos \theta}$$ or $$\frac{\cot^2 \theta + \cot \theta}{\cot^2 \theta - \cot \theta}$$ How do I continue?
 HW Helper P: 1,021 How did you end up with that? For the LHS: $$\frac{{1 + \tan \theta }}{{1 - \tan \theta }} = \frac{{1 + \frac{{\sin \theta }}{{\cos \theta }}}}{{1 - \frac{{\sin \theta }}{{\cos \theta }}}} = \frac{{\frac{{\cos \theta + \sin \theta }}{{\cos \theta }}}}{{\frac{{\cos \theta - \sin \theta }}{{\cos \theta }}}} = \frac{{\cos \theta + \sin \theta }}{{\cos \theta - \sin \theta }}$$ Now try the RHS
P: 786
 Quote by TD How did you end up with that? For the LHS: $$\frac{{1 + \tan \theta }}{{1 - \tan \theta }} = \frac{{1 + \frac{{\sin \theta }}{{\cos \theta }}}}{{1 - \frac{{\sin \theta }}{{\cos \theta }}}} = \frac{{\frac{{\cos \theta + \sin \theta }}{{\cos \theta }}}}{{\frac{{\cos \theta - \sin \theta }}{{\cos \theta }}}} = \frac{{\cos \theta + \sin \theta }}{{\cos \theta - \sin \theta }}$$ Now try the RHS
Silly me - I just multiplied out the numerator by the reciprocal of the denomenator instead of just canceling out the cosines. If you factor the top and bottom of my expression you end up with what your answer. If I do this using 1/cot = tan I end up with the RHS.

Don't I need to continue with the LHS until I get the right or vice versa?
 HW Helper P: 1,021 Well now you have the LHS, the easiest would be trying to get the same starting with the RHS, which will go more or less the same
 P: 786 Ah, I see. Thank you both of you.
 HW Helper P: 1,021 No problem

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