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JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Main · Mathematics · Trigonometric Equations

If a ^2 θ+b ^2 θ=c , then ^2 θ is equal to

If asin⁡2θ+bcos⁡2θ=ca \sin ^2 \theta+b \cos ^2 \theta=c, then tan⁡2θ\tan ^2 \theta is equal to

Options

  1. Ab−ca−c\frac{b-c}{a-c}
  2. Bc−ba−c\frac{c-b}{a-c}
  3. Ca−cb−c\frac{a-c}{b-c}
  4. Da−cc−b\frac{a-c}{c-b}

Correct answer

B. c−ba−c\frac{c-b}{a-c}

Step-by-step solution

asin⁡2θ+bcos⁡2θ=ca \sin ^2 \theta+b \cos ^2 \theta=c On dividing both sides by cos⁡2\thetaatan⁡2θ+b=csec⁡2θ⇒atan⁡2θ+b=c(1+tan⁡2θ)⇒atan⁡2θ+b=c+ctan⁡2θ⇒b=c+ctan⁡2θ−atan⁡2θ⇒(c−a)tan⁡2θ=(b−c)⇒tan⁡2θ=b−cc−a\cos ^2 \thetaa \tan ^2 \theta+b=c \sec ^2 \theta\Rightarrow \quad a \tan ^2 \theta+b=c\left(1+\tan ^2 \theta\right)\Rightarrow \quad a \tan ^2 \theta+b=c+c \tan ^2 \theta\Rightarrow \quad b=c+c \tan ^2 \theta-a \tan ^2 \theta\Rightarrow \quad(c-a) \tan ^2 \theta=(b-c)\Rightarrow \quad \tan ^2 \theta=\frac{b-c}{c-a} or c−ba−c\frac{c-b}{a-c}
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