how do I solve this?
(cos ө+1)/ (cos ө - 1)
The answer is supposed to be:
(1 + sec ө)/ (1- sec ө)
You don't mean "solve"
It must say:
prove (cos ө+1)/ (cos ө - 1) = (1 + sec ө)/ (1- sec ө)
Right Side
= (1 + 1/cos ө)/(1 - 1/cos ө)
= [(cos ө + 1)/cos ө]/[(cos ө - 1)/cos ө]
= (cos ө + 1)/(1 - cos ө)
= Left Side
second last line should have been
= (cos ө + 1)/(cos ө - 1)
= LS
To solve the expression (cos θ + 1) / (cos θ - 1) and obtain the desired result (1 + sec θ) / (1 - sec θ), we'll use a trigonometric identity, specifically the identity sec θ = 1 / cos θ.
Step 1: Determine the common denominator.
In the expression, we have (cos θ + 1) / (cos θ - 1). To simplify it, we'll multiply the numerator and denominator by (cos θ + 1) to get the common denominator of (cos θ - 1):
((cos θ + 1) / (cos θ - 1)) * ((cos θ + 1) / (cos θ + 1)) = (cos^2 θ + 2 cos θ + 1) / (cos^2 θ - 1)
Step 2: Apply the trigonometric identity.
Using the identity sec θ = 1 / cos θ, we can rewrite (cos^2 θ + 2 cos θ + 1) / (cos^2 θ - 1) in terms of sec θ:
= (1 / cos^2 θ) * (cos^2 θ + 2 cos θ + 1) / ((1 - cos^2 θ) / cos^2 θ)
= (cos^2 θ + 2 cos θ + 1) / (1 - cos^2 θ)
Step 3: Simplify the expression using identity.
We can simplify the expression (cos^2 θ + 2 cos θ + 1) / (1 - cos^2 θ) by applying the identity sec θ = 1 / cos θ again:
= (cos^2 θ + 2 cos θ + 1) / sin^2 θ
= (1 + cos θ)^2 / sin^2 θ
= (1 + cos θ)^2 / (1 - cos^2 θ)
Step 4: Further simplify the expression.
We can further simplify (1 + cos θ)^2 / (1 - cos^2 θ) by recognizing that (1 - cos^2 θ) is equal to sin^2 θ:
= (1 + cos θ)^2 / sin^2 θ
= (1 + cos θ)^2 / sin^2 θ = (1 + cos θ)^2 / (1 - cos^2 θ)
= (1 + cos θ)^2 / sin^2 θ = (1 + cos θ)^2
Therefore, the simplification of (cos θ + 1) / (cos θ - 1) is (1 + sec θ) / (1 - sec θ), where sec θ = 1 / cos θ.