how do I simplify this?
1. (Tan ө + cos ө)/ (sec ө + cot ө)
2. (Tan ө + cot ө)/ sec ө
To simplify these expressions, we can start by using trigonometric identities. Here's how you can simplify each expression:
1. (Tan ө + cos ө) / (sec ө + cot ө):
- First, let's express tan(ө) and cot(ө) in terms of sin(ө) and cos(ө):
tan(ө) = sin(ө) / cos(ө)
cot(ө) = cos(ө) / sin(ө)
- Next, let's express sec(ө) in terms of cos(ө) and cot(ө):
sec(ө) = 1 / cos(ө)
- Substituting these expressions into the original expression, we get:
((sin(ө) / cos(ө)) + cos(ө)) / (1 / cos(ө) + (cos(ө) / sin(ө)))
- Now, let's simplify the expression further. We can multiply the entire expression by sin(ө) * cos(ө) to get rid of the fractions:
(sin(ө) + cos^2(ө)) / (sin(ө) + cos(ө))
- Finally, we can factor out a common factor of (sin(ө) + cos(ө)):
(sin(ө) + cos(ө)) * (1 + cos(ө)) / (sin(ө) + cos(ө))
- The (sin(ө) + cos(ө)) terms cancel out, leaving us with the simplified expression of:
1 + cos(ө)
2. (Tan ө + cot ө) / sec ө:
- Following similar steps as in the previous expression, we can express tan(ө) and cot(ө) in terms of sin(ө) and cos(ө), and sec(ө) in terms of cos(ө) and cot(ө):
tan(ө) = sin(ө) / cos(ө)
cot(ө) = cos(ө) / sin(ө)
sec(ө) = 1 / cos(ө)
- Substitute these expressions into the original expression:
((sin(ө) / cos(ө)) + (cos(ө) / sin(ө))) / (1 / cos(ө))
- Simplify further by multiplying the entire expression by sin(ө) * cos(ө):
(sin^2(ө) + cos^2(ө)) / (sin(ө) * cos(ө)) / (1 / cos(ө))
- The sin^2(ө) + cos^2(ө) term simplifies to 1, so we are left with:
1 / (sin(ө) * cos(ө)) / (1 / cos(ө))
- Simplify the expression by canceling out the terms with cos(ө):
1 / sin(ө)
Therefore, the simplified expression is 1 / sin(ө), which is equivalent to csc(ө).
In summary:
1. (Tan ө + cos ө) / (sec ө + cot ө) simplifies to 1 + cos(ө).
2. (Tan ө + cot ө) / sec ө simplifies to 1 / sin(ө) or csc(ө).