Pregunta
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
\( \sin^{4} x - \cos^{4} x = -\cos(2x) \)
Solución
Calculate or simplify the expression \( \sin^{4}(x)-\cos^{4}(x) \).
Simplify the expression by following steps:
- step0: Solution:
\(\sin^{4}\left(x\right)-\cos^{4}\left(x\right)\)
- step1: Transform the expression:
\(\left(\sin^{2}\left(x\right)+\cos^{2}\left(x\right)\right)\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)\)
- step2: Transform the expression:
\(1\times \left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)\)
- step3: Multiply the terms:
\(\left(\sin\left(x\right)+\cos\left(x\right)\right)\left(\sin\left(x\right)-\cos\left(x\right)\right)\)
- step4: Transform the expression:
\(\sin^{2}\left(x\right)-\cos^{2}\left(x\right)\)
- step5: Transform the expression:
\(1-\cos^{2}\left(x\right)-\cos^{2}\left(x\right)\)
- step6: Subtract the terms:
\(1-2\cos^{2}\left(x\right)\)
- step7: Transform the expression:
\(1-1-\cos\left(2x\right)\)
- step8: Remove the opposites:
\(-\cos\left(2x\right)\)
The expression \( \sin ^{4} x - \cos ^{4} x \) simplifies to \( -\cos(2x) \).
Respondido por UpStudy AI y revisado por un tutor profesional
Como
error msg
Explicar 
Simplifique esta solución Bonus Knowledge
To simplify the expression \( \sin^{4} x - \cos^{4} x \), you can use the difference of squares formula. This gives you: \[ \sin^{4} x - \cos^{4} x = (\sin^{2} x + \cos^{2} x)(\sin^{2} x - \cos^{2} x). \] Since \( \sin^{2} x + \cos^{2} x = 1 \), the expression simplifies to: \[ \sin^{4} x - \cos^{4} x = 1 \cdot (\sin^{2} x - \cos^{2} x). \] Thus, the final simplified result is: \[ \sin^{4} x - \cos^{4} x = \sin^{2} x - \cos^{2} x. \] Now you're equipped to tackle any similar trigonometric expressions with confidence! Plus, remember that the identity \( \sin^{2} x - \cos^{2} x \) can further be expressed in terms of tangent for even deeper insights.
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde
Hazte Premium
