Hira S.

asked • 06/27/19# If Tan^2(t)-Sin^2(t)=Sin^a(t)/Cos^b(t), then the positive power a= ALSO the positive power b=

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!

## 1 Expert Answer

Rachel C. answered • 06/27/19

High School Math Teacher Looking To Help Your Student Succeed

First, we know that tan t = (sin t) / (cos t), so

sin^{2} t

tan^{2} t = _______

cos^{2} t

Rewriting the original questiona tan^{2} t - sin^{2} t is the same as

sin^{2} t

= _______ - sin^{2} t

cos^{2} t

We want to express this as a single fraction. Right now, we have the fractions,

sin^{2} t sin^{2} t

_______ - ________

cos^{2} t 1

In order to combine the fractions, we need a comm on denominator. Our common denominator would be cos^{2} t. But if we only change the denominator in the second fraction from 1 to cos^{2} 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 cos^{2 }t. To do this, we can multiply the fraction by cos^{2} t / cos^{2 }t since anything divided by itself is 1.

sin^{2} t cos^{2} t sin^{2 }t * cos^{2} t

________ * _________ = ________________

1 cos^{2} t cos^{2} t

sin^{2} t sin^{2} t * cos^{2} t

= ________ - _______________

cos^{2} t cos^{2} t

sin^{2} t - (sin^{2} t * cos^{2} t)

= __________________________

cos^{2}t

Both terms have sin^{2} t in common, so we can factor it out,

sin^{2} t * ( 1 - cos^{2} t)

= ______________________

cos^{2} t

The Pythagorean Identity tells us that cos^{2} t + sin^{2} t = 1. If we subtract cos^{2 }t from both sides, this gives us: sin^{2} t = 1 - cos^{2 }t

Therefore, we can substitute our ( 1 - cos^{2} t) with sin^{2 }t in the numerator, giving us

sin^{2} t * sin^{2} t

= __________________

cos^{2} t

= sin^{4} t

_____________

cos^{2} t

Therefore, a = 4 and b = 2

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Mark M.

Your first step is wrong. You changed the a to 2 and the b to 2.06/27/19