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My name is Donny and I'm prepping to take my ACTS in a few weeks. I got a bit stumped on these two problems and was hoping you could help (they are very similar, so I thought I could include them both, if that's alright).

I am asked to verify each of these identities
#1 csc^(2)x(1-cos^(2)x)= 1
#2 sec(theta)-sin(theta)*tan(theta)=cos(theta)

I was hoping you could just go through those two problems briefly; explain your answer and how you got them. They are similar problems, but different enough that I wanted to get the processes of both from you.

It is very useful to convert all terms to sin and cos.

#1 Since csc(x) is 1/sin(x) the cscē(x) is 1/sinē(x).
It is known that sinē(x) + cosē(x) = 1, so 1 - cosē(x) = sinē(x).
Putting these two in gives (1/sinē(x))sinē(x), and that reduces to 1.

#2 Put T in for theta and get sec(T) - sin(T)*tan(T).
Since sec(T) = 1/cos(T) and tan(T) = sin(T)/cos(T), we have 1/cos(T) - sin(T)sin(T)/cos(T).
Note that sin(T)sin(T) is sinē(T).
Also not the fractions can be combined.  This gives (1-sinē(T))/cos(T).

Since sinē(T) + cosē(T) = 1, 1 - sinē(T) = cosē(T), so this is cosē(T)/cos(T).
That reduces to cos(T), which is what's on the right.

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#### Scott A Wilson

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