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BITSAT 2021
Exam
BITSAT
Year
2021
Subject
Mathematics
Chapter
Trigonometric Equations

BITSAT 2021 · Mathematics · Trigonometric Equations

If y=(2 α)/(1+ α+ α) , then value of (1- α+ α)/(1+ α) is

If y=2sin⁡α1+cos⁡α+sin⁡αy=\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}, then value of 1−cos⁡α+sin⁡α1+sin⁡α\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha} is

Options

  1. Ay3\frac{y}{3}
  2. Byy
  3. C2y2 y
  4. D32y\frac{3}{2} y

Correct answer

B. yy

Step-by-step solution

1−cos⁡α+sin⁡α1+sin⁡α=1−cos⁡α+sin⁡α1+sin⁡α⋅1+cos⁡α+sin⁡α1+cos⁡α+sin⁡α=(1+sin⁡α)2−cos⁡2α(1+sin⁡α)(1+cos⁡α+sin⁡α)=(1+sin⁡2α+2sin⁡α)−(1−sin⁡2α)(1+sin⁡α)(1+cos⁡α+sin⁡α)=2sin⁡α1+cos⁡α+sin⁡α=y\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}=\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha} \cdot \frac{1+\cos \alpha+\sin \alpha}{1+\cos \alpha+\sin \alpha}=\frac{(1+\sin \alpha)^{2}-\cos ^{2} \alpha}{(1+\sin \alpha)(1+\cos \alpha+\sin \alpha)}=\frac{\left(1+\sin ^{2} \alpha+2 \sin \alpha\right)-\left(1-\sin ^{2} \alpha\right)}{(1+\sin \alpha)(1+\cos \alpha+\sin \alpha)}=\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=y
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