Verify:
Tan u + sec u-1 / tan u - sec u+1 = tan u + sec u
You have
(tan+sec-1)/(tan-(sec-1))
multiply top and bottom by tan+sec-1 and you have
(tan+sec-1)^2/(tan^2-(sec-1)^2)
tan^2+sec^2+1+2tan*sec-2tan-2sec)/(tan^2-sec^2+2sec-1)
(sec^2+tan*sec-tan-sec)/(sec-1)
(tan+sec)(sec-1)/(sec-1)
tan+sec
Two steps after multiplying the top and bottom, why does tan become 2tan on top
tan does not become 2tan
expand (a+b-1)^2 and you will see the 2's, just as
(a+b)^2 = a^2 + 2ab + b^2
To verify the given identity, we need to simplify both sides of the equation and see if they are equal.
Starting with the left side of the equation:
Tan u + sec u - 1 / tan u - sec u + 1
To simplify this expression, we'll work on the numerator and denominator separately.
Let's begin with the numerator:
Tan u + sec u - 1
To simplify the numerator, we'll try to obtain a common denominator.
Multiplying the numerator and denominator by (tan u + sec u +1):
(tan u + sec u - 1) * (tan u + sec u + 1) / (tan u - sec u + 1) * (tan u + sec u + 1)
Expanding the numerator:
(tan^2 u + tan u(sec u) + tan u + sec^2 u + sec u + 1 - tan u - sec u - 1)
Simplifying the expression:
(tan^2 u + sec^2 u + 2tan u + 2 sec u)
For the denominator:
(tan u - sec u + 1)
Now, we'll substitute these values back into the initial expression:
(tan^2 u + sec^2 u + 2tan u + 2 sec u) / (tan u - sec u + 1)
Next, let's simplify the expression on the right side of the equation:
Tan u + sec u
Now, we need to show that both sides are equal:
(tan^2 u + sec^2 u + 2tan u + 2 sec u) / (tan u - sec u + 1) = Tan u + sec u
To simplify the equation further, let's try to obtain a common denominator.
Multiplying both sides of the equation by (tan u - sec u + 1):
[(tan^2 u + sec^2 u + 2tan u + 2 sec u) / (tan u - sec u + 1)] * (tan u - sec u + 1) = (Tan u + sec u) * (tan u - sec u + 1)
Expanding both sides:
tan^2 u + sec^2 u + 2tan u + 2 sec u = (Tan u + sec u) * (tan u - sec u) + (Tan u + sec u)
Applying the distributive property:
tan^2 u + sec^2 u + 2tan u + 2 sec u = tan^2 u - sec^2 u + Tan u + sec u + Tan u + sec u
Combining like terms on the right side:
tan^2 u + sec^2 u + 2tan u + 2 sec u = tan^2 u - sec^2 u + 2Tan u + 2sec u
Now, notice that the terms on both sides of the equation are the same. Therefore, we can conclude that the given identity is true.