Question
a) \( \sin ^{4} x+\cos ^{2} x \cdot \sin ^{2} x-\sin ^{2} x \)
Ask by Peters Campos. in Turkey
Nov 25,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \sin^4 x + \cos^2 x \cdot \sin^2 x - \sin^2 x = 0 \)
Solution
Calculate or simplify the expression \( \sin^4 x + \cos^2 x \cdot \sin^2 x - \sin^2 x \).
Simplify the expression by following steps:
- step0: Solution:
\(\sin^{4}\left(x\right)+\cos^{2}\left(x\right)\sin^{2}\left(x\right)-\sin^{2}\left(x\right)\)
- step1: Multiply the terms:
\(\sin^{4}\left(x\right)+\left(\cos\left(x\right)\sin\left(x\right)\right)^{2}-\sin^{2}\left(x\right)\)
- step2: Transform the expression:
\(\sin^{2}\left(x\right)\left(\sin^{2}\left(x\right)+\cos^{2}\left(x\right)-1\right)\)
- step3: Transform the expression:
\(\sin^{2}\left(x\right)\left(1-1\right)\)
- step4: Subtract the terms:
\(\sin^{2}\left(x\right)\times 0\)
- step5: Multiply:
\(0\)
The expression \( \sin^4 x + \cos^2 x \cdot \sin^2 x - \sin^2 x \) simplifies to 0.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
First, let's simplify the expression step by step: 1. Start with the original expression: \[ \sin^4 x + \cos^2 x \cdot \sin^2 x - \sin^2 x. \] 2. Notice that \( \cos^2 x = 1 - \sin^2 x \). We can substitute this into our expression: \[ \sin^4 x + (1 - \sin^2 x) \cdot \sin^2 x - \sin^2 x. \] 3. Distributing \( \sin^2 x \) gives: \[ \sin^4 x + \sin^2 x - \sin^4 x - \sin^2 x. \] 4. Combining like terms, we see that the \(\sin^4 x\) terms and the \(\sin^2 x\) terms cancel out: \[ 0. \] Thus, the simplified form of the expression is \(0\).
