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kengeta
how do i prove : tan^2 θ cos^2 θ + cos^2 θ = 1
well my first attempt to prove this is true is to take the left hand side and try to show the right hand side the left hand side has both terms with the factor cos^2(theta) start by factoring cos^2(theta) out from both terms this will get us closer to writting as 1 term just as the right hand side is (one term)
let me know if you still don't know where to go after that
do you know hot to factor the expression ax+x?
I will show you how to factor ax+x then you should be able to factor tan^2(theta)*cos^2(theta)+cos^2(theta) so ax+x I see there is a x in both terms |dw:1407522006591:dw| so i will factor that x out like so \[ax+1x=x(a+1)\]
you have \[\tan^2(\theta) \cdot \cos^2(\theta)+\cos^2(\theta) \cdot 1 \] \[\text{ do you see that there is a } cos^2(\theta) \text{ in both terms ?}\]
can you factor that out using what you know about multiplication and division like i did above?
also this way i'm asking you to approach it is not the only approach to take
ok if you are having trouble factoring the cos^2(theta) out how about we try this another way do you recall tan is sin/cos?
\[\frac{\sin^2(\theta)}{\cos^2(\theta)} \cos^2(\theta)+\cos^2(\theta)\] in the first term there, do you see anything that cancels?
right so what does that leave us with?
\[\frac{\sin^2(\theta)}{\cancel{\cos^2(\theta)}}\cancel{\cos^2(\theta)}+\cos^2(\theta)\] what does this give us?
sin^2(theta)+cos^2(theta)
you need to recall some trig identities for these trigonometric proofs
This one of the most basic ones It actually has a name
I will give you a hint: Pythagorean Identity
okay i get it now thanks soooo much
Way 1: \[\tan^2(\theta) \cdot \cos^2(\theta)+\cos^2(\theta) \cdot 1= \\ \cos^2(\theta)(\tan^2(\theta)+1)= \\ \cos^2(\theta) \cdot \sec^2(\theta)= \\ 1\] Way 2: \[\tan^2(\theta) \cos^2(\theta)+\cos^2(\theta)= \\ \frac{\sin^2(\theta)}{\cos^2(\theta)} \cos^2(\theta)+\cos^2(\theta) = \\ \sin^2(\theta) +\cos^2(\theta)= \\ 1\] These are the two ways that I can think of... This does not mean they are the only ways.
You don't have to do both ways. Just one way.
I used a Pythagorean identity in both ways.
way 1 was the way I was trying to get you to go about it the first time around
I don't think one way is more harder than the other But you will have to review some algebra especially if you don't remember how to factor. Factoring will come up again.