Question

`(\sec\left(x\right))/(\cos\left(x\right))-(\sin\left(x\right))/(\csc\left(x\right)\cos^(2)\left(x\right))`Use the basic identities to change the expression to one involving only sines and cosines. Then simplify to a basic trig function.

Answer 1

1

Explanation:

First, convert all the secants and cosecants to cosine and sine, respectively. Recall that
`csc(x)=1/sin(x)` and
`sec(x)=1/cos(x)`.

Thus:

`(sec(x))/(cos(x)) -(sin(x))/(csc(x)cos^2(x))`

`=((1)/(cos(x)) )/(cos(x)) -(sin(x))/((1)/(sin(x))cos^2(x) )`

Let's do the first part first: (Recall how to divide fractions)

`((1)/(cos(x)) )/(cos(x))=(1)/(cos(x)) \cdot (1)/(cos(x))=(1)/(cos^2(x))`

For the second term:

`(sin(x))/((cos^2(x))/(sin(x)) ) =(sin(x))/(1) \cdot(sin(x))/(cos^2(x))=(sin^2(x))/(cos^2(x))`

So, all together: (same denominator; combine terms)

`(1)/(cos^2(x))-(sin^2(x))/(cos^2(x))=(1-sin^2(x))/(cos^2(x))`

Note the numerator; it can be derived from the Pythagorean Identity:

`sin^2(x)+cos^2(x)=1; cos^2(x)=1-sin^2(x)`

Thus, we can substitute the numerator:

`(1-sin^2(x))/(cos^2(x))=(cos^2(x))/(cos^2(x))=1`

Everything simplifies to 1.