If \cos \theta + \sec \theta = k, then what is the value of \sin^{2}\theta - \tan^{2}\theta?

Explanation

Rewrite the expression: \sin^2\theta - \tan^2\theta = (1-\cos^2\theta) - (\sec^2\theta - 1) = 2 - (\cos^2\theta + \sec^2\theta). We know k^2 = (\cos\theta+\sec\theta)^2 = \cos^2\theta + \sec^2\theta + 2, which implies \cos^2\theta + \sec^2\theta = k^2 - 2. Thus, 2 - (k^2 - 2) = 4 - k^2.

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