If 7 \sin^2 \theta + 3 \cos^2 \theta = 4, 0^\circ < \theta < 90^\circ, then the value of (\tan^2 2 \theta + \cosec^2 2 \theta) is:

Question 126

If $$7 \sin^2 \theta + 3 \cos^2 \theta = 4, 0^\circ < \theta < 90^\circ$$, then the value of $$(\tan^2 2 \theta + \cosec^2 2 \theta)$$ is:

Solution

Expression : $$7 \sin^2 \theta + 3 \cos^2 \theta = 4, 0^\circ < \theta < 90^\circ$$

=> $$7sin^2\theta+3(1-sin^2\theta)=4$$

=> $$4sin^2\theta=1$$

=> $$sin\theta=\sqrt{\frac{1}{4}}$$

=> $$\theta=sin^{-1}(\frac{1}{2})=30^\circ$$

To find : $$(\tan^2 2 \theta + \cosec^2 2 \theta)$$

= $$tan^2(60^\circ)+cosec^2(60^\circ)$$

= $$3+\frac{4}{3}=\frac{13}{3}$$

=> Ans - (C)

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