Question
upstudy study bank question image url

2. \( \frac{2 \sin ^{2} x}{2 \tan x-\sin 2 x}=\frac{1}{\tan x} \)

Ask by Gardner Harris. in South Africa
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The equation is an identity, meaning it holds true for all real numbers \( x \) except where \( \sin x = 0 \) or \( \cos x = 0 \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Bonus Knowledge

To solve the equation \( \frac{2 \sin ^{2} x}{2 \tan x-\sin 2 x}=\frac{1}{\tan x} \), we begin by rewriting the terms in terms of sine and cosine. Recall that: - \( \tan x = \frac{\sin x}{\cos x} \) - \( \sin 2x = 2 \sin x \cos x \) Now, substituting these into the equation gives: \[ \frac{2 \sin^2 x}{2 \frac{\sin x}{\cos x} - 2 \sin x \cos x} = \frac{1}{\frac{\sin x}{\cos x}} \] Simplifying the left side: \[ \frac{2 \sin^2 x}{\frac{2 \sin x}{\cos x} - 2 \sin x \cos x} = \frac{1}{\frac{\sin x}{\cos x}} \] This reduces to: \[ \frac{2 \sin^2 x \cos x}{2 \sin x - 2 \sin x \cos^2 x} = \frac{\cos x}{\sin x} \] Cross-multiplying gives: \[ 2 \sin^2 x \cos^2 x = \cos x (2 \sin x - 2 \sin x \cos^2 x) \] Dividing both sides by \(\cos x\) (assuming \(\cos x \neq 0\)) yields: \[ 2 \sin^2 x \cos x = 2 \sin x - 2 \sin x \cos^2 x \] Rearranging gives us: \[ 2 \sin^2 x \cos x + 2 \sin x \cos^2 x = 2 \sin x \] Factoring out \(2 \sin x\): \[ 2 \sin x (\sin x \cos x + \cos^2 x) = 2 \sin x \] Assuming \(\sin x \neq 0\), we can divide both sides by \(2 \sin x\): \[ \sin x \cos x + \cos^2 x = 1 \] Here, we can use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to rewrite the equation: \[ \sin x \cos x + (1 - \sin^2 x) = 1 \] This simplifies to: \[ \sin x \cos x = \sin^2 x \] Dividing both sides by \(\sin x\) (assuming \(\sin x \neq 0\)), we find: \[ \cos x = \sin x \] This results in: \[ \tan x = 1 \] The solutions for this equation are: \[ x = \frac{\pi}{4} + k\pi \quad \text{for } k \in \mathbb{Z} \] Thus, the final answer for the given equation is: \[ x = \frac{\pi}{4} + k\pi \quad (k \in \mathbb{Z}) \]

Related Questions

Latest Trigonometry Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy