- #1
cscott
- 782
- 1
Can someone please help me establish this identity?
[tex]\cos \theta (\tan \theta + \cot \theta) = \csc \theta[/tex]
[tex]\cos \theta (\tan \theta + \cot \theta) = \csc \theta[/tex]
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
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
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.
There are an infinite number of trigonometric identities. Some common ones include the Pythagorean identities, double angle identities, and sum and difference identities.
Trigonometric identities are useful because they allow us to simplify complex trigonometric expressions, solve equations, and prove other mathematical theorems.
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.
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.