Answer:
1
Step-by-step explanation:
First, convert all the secants and cosecants to cosine and sine, respectively. Recall that [tex]csc(x)=1/sin(x)[/tex] and [tex]sec(x)=1/cos(x)[/tex].
Thus:
[tex]\frac{sec(x)}{cos(x)} -\frac{sin(x)}{csc(x)cos^2(x)}[/tex]
[tex]=\frac{\frac{1}{cos(x)} }{cos(x)} -\frac{sin(x)}{\frac{1}{sin(x)}cos^2(x) }[/tex]
Let's do the first part first: (Recall how to divide fractions)
[tex]\frac{\frac{1}{cos(x)} }{cos(x)}=\frac{1}{cos(x)} \cdot \frac{1}{cos(x)}=\frac{1}{cos^2(x)}[/tex]
For the second term:
[tex]\frac{sin(x)}{\frac{cos^2(x)}{sin(x)} } =\frac{sin(x)}{1} \cdot\frac{sin(x)}{cos^2(x)}=\frac{sin^2(x)}{cos^2(x)}[/tex]
So, all together: (same denominator; combine terms)
[tex]\frac{1}{cos^2(x)}-\frac{sin^2(x)}{cos^2(x)}=\frac{1-sin^2(x)}{cos^2(x)}[/tex]
Note the numerator; it can be derived from the Pythagorean Identity:
[tex]sin^2(x)+cos^2(x)=1; cos^2(x)=1-sin^2(x)[/tex]
Thus, we can substitute the numerator:
[tex]\frac{1-sin^2(x)}{cos^2(x)}=\frac{cos^2(x)}{cos^2(x)}=1[/tex]
Everything simplifies to 1.
