Trig Identity: Solving \cos \theta (\tan \theta + \cot \theta) = \csc \theta

In summary, the conversation is about proving a trigonometric identity and using various substitutions and algebraic manipulations to simplify the expressions. The main focus is on expressing tangent and cotangent in terms of sine and cosine and using their LCD to transform the expressions into a well-known identity. One person is struggling with the proofs and is asking for help, while the other is providing guidance and explaining the steps.
  • #1
cscott
782
1
Can someone please help me establish this identity?

[tex]\cos \theta (\tan \theta + \cot \theta) = \csc \theta[/tex]
 
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  • #2
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
 
  • #3
irony of truth said:
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!
 
  • #4
I can't get this one either:

[tex]\frac{1 + \tan \theta}{1 - \tan \theta} = \frac{\cot \theta + 1}{\cot \theta - 1}[/tex]

I'm so bad at proofs :frown:
 
  • #5
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.
 
  • #6
Or, if that doesn't work for you, substitute tan by sin/cos and cot by cos/sin, then simplify the expressions :smile:

Try, if you get stuck, show us!
 
  • #7
I end up with

[tex]\frac{\cos^2 \theta + \sin \theta \cos \theta}{\cos^2 \theta - \sin \theta \cos \theta}[/tex]

or

[tex]\frac{\cot^2 \theta + \cot \theta}{\cot^2 \theta - \cot \theta}[/tex]

How do I continue?
 
  • #8
How did you end up with that?

For the LHS:

[tex]\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 }}[/tex]

Now try the RHS :smile:
 
  • #9
TD said:
How did you end up with that?

For the LHS:

[tex]\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 }}[/tex]

Now try the RHS :smile:

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?
 
  • #10
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 :smile:
 
  • #11
Ah, I see. Thank you both of you.
 
  • #12
No problem :smile:
 

What is a trigonometric identity?

A trigonometric identity is an equation that is true for all values of the variables involved. It shows the relationship between different trigonometric functions and is used to simplify expressions involving trigonometric functions.

How many trigonometric identities are there?

There are an infinite number of trigonometric identities. Some common ones include the Pythagorean identities, double angle identities, and sum and difference identities.

Why are trigonometric identities useful?

Trigonometric identities are useful because they allow us to simplify complex trigonometric expressions, solve equations, and prove other mathematical theorems.

How do I prove a trigonometric identity?

To prove a trigonometric identity, you need to manipulate one side of the equation using algebraic and trigonometric properties until it is equivalent to the other side. This can involve using common trigonometric identities or converting between different forms of trigonometric functions.

What are some real-life applications of trigonometric identities?

Trigonometric identities are used in many fields, including engineering, physics, and navigation. They can be used to calculate distances, angles, and other measurements in real-world scenarios involving triangles and circular motion.

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