Question - Proving a Trigonometric Identity
Solution:
The given equation is a trigonometric identity that we need to prove. Let's start by working with the left-hand side (LHS) of the equation and try to transform it into the right-hand side (RHS).The LHS of the equation is:\[ \frac{2 \tan \theta}{1 + \tan^2 \theta} \]We know that $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$, and we can use this to rewrite the LHS:\[ \frac{2 \frac{\sin \theta}{\cos \theta}}{1 + \left(\frac{\sin \theta}{\cos \theta}\right)^2} \]Simplify it further by combining the terms:\[ \frac{2\sin \theta}{\cos \theta} \times \frac{\cos^2 \theta}{\cos^2 \theta + \sin^2 \theta} \]Since $$\cos^2 \theta + \sin^2 \theta = 1$$ (which is another trigonometric identity), we can simplify:\[ \frac{2\sin \theta}{\cos \theta} \times \frac{\cos^2 \theta}{1} \]\[ 2\sin \theta \times \cos \theta \]Which is the RHS of the equation:\[ 2\sin \theta \cos \theta \]Therefore, we have proved that the LHS is equal to the RHS:\[ \frac{2 \tan \theta}{1 + \tan^2 \theta} = 2 \sin \theta \cos \theta \]And this confirms that the initial statement is a valid trigonometric identity.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com