Answer :
To simplify the expression [tex]\(\sec(\theta) - \cos(\theta)\)[/tex], let's proceed step-by-step.
1. Rewrite [tex]\(\sec(\theta)\)[/tex] in terms of [tex]\(\cos(\theta)\)[/tex]:
By definition, [tex]\(\sec(\theta)\)[/tex] is the reciprocal of [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} \][/tex]
2. Substitute [tex]\(\sec(\theta)\)[/tex] into the original expression:
Replace [tex]\(\sec(\theta)\)[/tex] in the expression [tex]\(\sec(\theta) - \cos(\theta)\)[/tex]:
[tex]\[ \sec(\theta) - \cos(\theta) = \frac{1}{\cos(\theta)} - \cos(\theta) \][/tex]
3. Combine the terms by finding a common denominator:
To subtract these fractions, we need a common denominator, which in this case is [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \frac{1}{\cos(\theta)} - \cos(\theta) = \frac{1}{\cos(\theta)} - \frac{\cos^2(\theta)}{\cos(\theta)} = \frac{1 - \cos^2(\theta)}{\cos(\theta)} \][/tex]
4. Use the Pythagorean identity:
According to the Pythagorean identity, [tex]\(1 - \cos^2(\theta) = \sin^2(\theta)\)[/tex]. Substitute this identity into the expression:
[tex]\[ \frac{1 - \cos^2(\theta)}{\cos(\theta)} = \frac{\sin^2(\theta)}{\cos(\theta)} \][/tex]
Therefore, the simplified form of the expression [tex]\(\sec(\theta) - \cos(\theta)\)[/tex] is:
[tex]\[ \boxed{\frac{\sin^2(\theta)}{\cos(\theta)}} \][/tex]
1. Rewrite [tex]\(\sec(\theta)\)[/tex] in terms of [tex]\(\cos(\theta)\)[/tex]:
By definition, [tex]\(\sec(\theta)\)[/tex] is the reciprocal of [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} \][/tex]
2. Substitute [tex]\(\sec(\theta)\)[/tex] into the original expression:
Replace [tex]\(\sec(\theta)\)[/tex] in the expression [tex]\(\sec(\theta) - \cos(\theta)\)[/tex]:
[tex]\[ \sec(\theta) - \cos(\theta) = \frac{1}{\cos(\theta)} - \cos(\theta) \][/tex]
3. Combine the terms by finding a common denominator:
To subtract these fractions, we need a common denominator, which in this case is [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \frac{1}{\cos(\theta)} - \cos(\theta) = \frac{1}{\cos(\theta)} - \frac{\cos^2(\theta)}{\cos(\theta)} = \frac{1 - \cos^2(\theta)}{\cos(\theta)} \][/tex]
4. Use the Pythagorean identity:
According to the Pythagorean identity, [tex]\(1 - \cos^2(\theta) = \sin^2(\theta)\)[/tex]. Substitute this identity into the expression:
[tex]\[ \frac{1 - \cos^2(\theta)}{\cos(\theta)} = \frac{\sin^2(\theta)}{\cos(\theta)} \][/tex]
Therefore, the simplified form of the expression [tex]\(\sec(\theta) - \cos(\theta)\)[/tex] is:
[tex]\[ \boxed{\frac{\sin^2(\theta)}{\cos(\theta)}} \][/tex]
