Answer
$$\tan^{2}\theta$$
Work Step by Step
Use the Pythagorean Identity:
$$1+\tan^{2}\theta=\sec^{2}\theta$$
Rearrange terms:
$$\sec^{2}\theta-1=\tan^{2}\theta$$
Alternatively, replace $\sec{\theta}$ with $\frac{1}{\cos\theta}$
$$\sec^{2}\theta-1= \frac{1}{cos^{2}\theta}-1$$
Common denominator:
$$\sec^{2}\theta-1= \frac{1-cos^{2}\theta}{cos^{2}\theta}$$
By identity, $sin^{2}\theta+cos^{2}\theta=1$ (subtract $cos^{2}\theta$ on both sides)
Therefore, replace 1-$cos^{2}\theta$ with $sin^{2}\theta$
$$\sec^{2}\theta-1=\frac{sin^{2}\theta}{cos^{2}\theta}$$
By identity: $\frac{sin\theta}{cos\theta}=tan\theta$
Hence,
$$\sec^{2}\theta-1=tan^{2}\theta$$
