If 6+8 \tan \theta=\sec \theta and 8-6 \tan \theta=k \sec \theta, then what is the value of k^2?

Explanation

Squaring and adding both equations: (6+8\tan\theta)^2 + (8-6\tan\theta)^2 = \sec^2\theta + k^2\sec^2\theta. Expanding yields 36 + 64\tan^2\theta + 64 + 36\tan^2\theta = (1+k^2)\sec^2\theta, so 100(1+\tan^2\theta) = (1+k^2)\sec^2\theta. Since 1+\tan^2\theta = \sec^2\theta, we get 1+k^2 = 100, leading to k^2 = 99.

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