for (sec x -1)(sec x + 1) = tan^(2) x
so far I got up to:
(sin^(2)x / cos x) (-sin^(2)x / cos x)
what would the next step be?
steps too please
LS = sec^2 x - 1 , (did you notice the difference of squares pattern ?)
= tan^2 x , by definition
= RS
I don't get how you got to sec^2x - 1
where did the 1 come from
To simplify the expression further, you can use the trigonometric Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. In this case, you already have sin^2(x) in the numerator and cos x in the denominator.
Let's rewrite the expression keeping this identity in mind:
(sin^2(x) / cos(x)) * (-sin^2(x) / cos(x))
Next, you can simplify the expression by canceling out the common factors in the numerator and denominator. In this case, the common factor is sin^2(x).
(-sin^2(x) * sin^2(x)) / (cos(x) * cos(x))
Now, simplify the denominator further by using the squared trignometric identity, cos^2(x) = 1 - sin^2(x):
(-sin^2(x) * sin^2(x)) / ((1 - sin^2(x)) * cos(x))
You can now simplify the numerator by multiplying through:
(-sin^4(x)) / ((1 - sin^2(x)) * cos(x))
Lastly, you can use an identity for cos(x) to further simplify the expression, cos(x) = sqrt(1 - sin^2(x)):
(-sin^4(x)) / ((1 - sin^2(x)) * sqrt(1 - sin^2(x)))
So, the simplified form of (sec x - 1)(sec x + 1) is:
(-sin^4(x)) / ((1 - sin^2(x)) * sqrt(1 - sin^2(x)))
Note: This is just one possible method of simplifying the expression. There may be alternative approaches as well.