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So, I know the first two steps
sin^2(t)/cos^2(t) -sin^2(t)
sin^2(t)-sin^2(t)cos^2(t)/(cos^2(t)
I have no idea what to do after this? I looked and it said to factor but I have no idea why you factor sin^2(t)???
^This is me just simplifying the left side.
Please show all steps so I can understand how to get the answer! Thank you!
Solution
First, we know that tan t = (sin t) / (cos t), so
sin2 t
tan2 t = _______
cos2 t
Rewriting the original questiona tan2 t – sin2 t is the same as
sin2 t
= _______ – sin2 t
cos2 t
We want to express this as a single fraction. Right now, we have the fractions,
sin2 t sin2 t
_______ – ________
cos2 t 1
In order to combine the fractions, we need a comm on denominator. Our common denominator would be cos2 t. But if we only change the denominator in the second fraction from 1 to cos2 t, we’ve changed the problem. The only value you can multiply the fraction by and not make a new problem is 1. But we still want the denominator to equal cos2 t. To do this, we can multiply the fraction by cos2 t / cos2 t since anything divided by itself is 1.
sin2 t cos2 t sin2 t * cos2 t
________ * _________ = ________________
1 cos2 t cos2 t
sin2 t sin2 t * cos2 t
= ________ – _______________
cos2 t cos2 t
sin2 t – (sin2 t * cos2 t)
= __________________________
cos2t
Both terms have sin2 t in common, so we can factor it out,
sin2 t * ( 1 – cos2 t)
= ______________________
cos2 t
The Pythagorean Identity tells us that cos2 t + sin2 t = 1. If we subtract cos2 t from both sides, this gives us: sin2 t = 1 – cos2 t
Therefore, we can substitute our ( 1 – cos2 t) with sin2 t in the numerator, giving us
sin2 t * sin2 t
= __________________
cos2 t
= sin4 t
_____________
cos2 t
Therefore, a = 4 and b = 2