证明：$( \sin \theta + \csc \theta )^{2} + ( \cos \theta + \sec \theta )^{2} = 7 + \tan^{2} \theta + \cot^{2} \theta$。

已知： $( \sin \theta + \csc \theta )^{2} + ( \cos \theta + \sec \theta )^{2} = 7 + \tan^{2} \theta + \cot^{2} \theta$。

要求：证明左边 = 右边。

左边 = $( \sin \theta + \csc \theta )^{2} + ( \cos \theta + \sec \theta )^{2}$

$= \sin^{2} \theta + \csc^{2} \theta + 2 \sin \theta \csc \theta + \cos^{2} \theta + \sec^{2} \theta + 2 \cos \theta \sec \theta$

$= \sin^{2} \theta + \cos^{2} \theta + 2 \sin \theta \csc \theta + \sec^{2} \theta + \csc^{2} \theta + 2 \cos \theta \sec \theta$

$= 1 + 2 + 2 + \sec^{2} \theta + \csc^{2} \theta$ [$\because \sin^{2}\theta + \cos^{2}\theta = 1$, $\sin \theta = \frac{1}{\csc \theta}$ 且 $\cos \theta = \frac{1}{\sec \theta}$]

$= 5 + 1 + \tan^{2} \theta + 1 + \cot^{2} \theta$ [$\sec^{2} \theta = 1 + \tan^{2} \theta$ 且 $\csc^{2} \theta = 1 + \cot^{2} \theta$]

$= 7 + \tan^{2} \theta + \cot^{2} \theta$

$= 右边$

因此证明左边 = 右边。