If $\frac{sin\theta + cos \theta}{sin\theta - cos \theta}=3,$ then the value of $sin^4 \theta - cos^4 \theta $ is equal to :
Answer & explanation
Correct answer: option 3
\(\frac{sin θ + cos θ}{sin θ - cos θ }\) = 3
sin θ + cos θ = 3sin θ - 3cos θ
2sin θ = 4cos θ
tan θ = 2
Now,
sin4 θ - cos4 θ
= (sin² θ + cos² θ) . ( sin² θ - cos² θ )
= sin² θ - cos² θ { sin² θ + cos² θ= 1 }
= - ( cos² θ - sin² θ )
= - cos2θ { cos² θ - sin² θ = cos2θ }
= - \(\frac{1- tan² θ}{1+ tan² θ }\)
= - \(\frac{1- 2²}{1+ 2² }\)
= \(\frac{3}{5}\)