Answer:
[tex]\frac{\sin^2(\theta)}{\cos(\theta)}[/tex]
Step-by-step explanation:
[tex]\sec(\theta)-\frac{1}{\sec(\theta)}[/tex]
Write first term as a fraction:
[tex]\frac{\sec(\theta)}{1}-\frac{1}{\sec(\theta)}[/tex]
Multiply first fraction by [tex]1=\frac{\sec(\theta)}{\sec(\theta)}[/tex]:
[tex]\frac{\sec^2(\theta)-1}{\sec(\theta)}[/tex]
Use Pythagoren Identity, [tex]\tan^2(\theta)+1=\sec^2(\theta)[/tex]:
[tex]\frac{\tan^2(\theta)}{\sec(\theta)}[/tex]
Rewrite using quotient identity, [tex]\frac{\sin(\theta)}{\cos(\theta)}=\tan(\theta)[/tex], and recirpocal identity, [tex]\frac{1}{\cos(\theta)}=\sec(\theta)[/tex]:
[tex]\frac{\frac{\sin^2(\theta)}{\cos^2(\theta)}}{\frac{1}{\cos(\theta)}}[/tex]
A factor of [tex]\frac{1}{\cos(\theta)}[/tex] cancels:
[tex]\frac{\sin^2(\theta)}{\cos(\theta)}[/tex]
