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Question 67
If $$(\cos^{2} \theta - 1)(1 + \tan^{2} \theta) + 2 \tan^2 \theta = 1, 0^\circ < \theta < 90^\circ$$ then $$\theta$$ is:
Solution
$$(\cos^{2} \theta - 1)(1 + \tan^{2} \theta) + 2 \tan^2 \theta = 1$$
 $$(-\sin^{2} \theta )(sec^2 \theta) + 2 \tan^2 \theta = 1$$
$$\frac{-\sin^{2} \theta}{cos^2 \theta} + 2 \tan^2 \theta = 1$$
$$-\tan^2 \theta + 2 \tan^2 \theta = 1$$
$$\tan^2 \theta = 1$$
For 0^\circ < \theta < 90^\circ$$,
$$\theta = 45\degree$$
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