<s>[INST] How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $ How to simplify the following expression :
$$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
 [/INST] Write $1$ in the numerator as : $$\sec^2(\theta) - \tan^2(\theta)$$
$$\frac{(\sec\theta -\tan\theta)^2+\sec^2\theta - \tan^2\theta}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
$$\frac{(\sec\theta -\tan\theta)^2+(\sec\theta - \tan\theta)(\sec\theta + \tan\theta)}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
$$\frac{(\sec\theta - \tan\theta)(\sec\theta - \tan\theta + \sec\theta + \tan\theta)}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
$$\frac{(\sec\theta - \tan\theta)(2 \sec\theta)}{\csc\theta(\sec\theta - \tan\theta)}$$
$$2 \tan\theta$$
Hence the simplified result is: $$2 \tan\theta$$
Hope the answer is clear !
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