Question
a) \( \frac{\sin ^{2} x+\sin ^{2} x \cos x}{1+\cos x} \)
Ask by Valdez Frank. in South Africa
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to sin²x.
Solution
We begin with the expression:
(sin²x + sin²x⋅cos x) / (1 + cos x)
Step 1: Factor the numerator. Notice that sin²x is common in both terms:
sin²x (1 + cos x) / (1 + cos x)
Step 2: Cancel the common factor (1 + cos x) in the numerator and denominator (provided 1 + cos x ≠ 0):
= sin²x
Thus, the expression simplifies to sin²x.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To simplify the expression \( \frac{\sin^{2} x + \sin^{2} x \cos x}{1 + \cos x} \), we can factor out \( \sin^{2} x \) from the numerator: \[ = \frac{\sin^{2} x (1 + \cos x)}{1 + \cos x} \] Next, as long as \( 1 + \cos x \neq 0 \) (which happens when \( x \) is an odd multiple of \( \pi \)), we can cancel \( 1 + \cos x \): \[ = \sin^{2} x \] Thus, for \( x \) not equal to those odd multiples, the expression simplifies to: \[ \sin^{2} x \] For the special cases when \( x = (2n+1)\pi \) where \( n \) is an integer, we would need to evaluate \( \frac{\sin^{2} x}{0} \), which tends to be undefined. So the simplified form is valid except at these specific points!
