Whovian said:Just write this as [itex]\displaystyle\dfrac{ 1}{\sin\left(2\cdot x\right)} -\dfrac{\cos\left(2\cdot x\right)}{ \sin\left(2\cdot x\right)}=\dfrac{ \sin\left(x\right)}{\cos\left(x\right)}[/itex] while remembering that [itex]\sin^2\left(x\right)+\cos^2\left(x\right)=1[/itex]. Then apply the standard trig identities.
Oh, and, is this really calculus?
The odd trig identity that often needs proof is sin(x)cos(y) - cos(x)sin(y) = sin(x - y).
Proving this identity is important because it is a fundamental property of trigonometric functions and is useful in solving various trigonometric equations and simplifying expressions.
To prove this identity, you can use the trigonometric identities for sin(x-y) and cos(x-y), as well as the properties of even and odd functions. You can also use geometric proofs or mathematical induction.
Yes, for example, we can use the trigonometric identities sin(x-y) = sin(x)cos(y) - cos(x)sin(y) and cos(x-y) = cos(x)cos(y) + sin(x)sin(y) to prove that sin(x)cos(y) - cos(x)sin(y) = sin(x - y). This can be seen by substituting x-y for x in the first identity and y for y in the second identity and then combining the two equations.
The significance of even and odd functions is that they allow us to simplify expressions and prove identities by using symmetry and properties that are true for only even or odd functions. In this case, the odd function property allows us to prove the odd trig identity by breaking it into simpler even and odd functions.