Question

I'm really struggling with establishing identities and I would seriously love some help.

Here are two questions I need answered:
https://i.imgur.com/BnFBQ33.png
and
https://i.imgur.com/nK2VeCH.png

I'm not sure where to start with either of these, and I really would like to know how to solve them and what the answers are. Any advice would be helpful!

Thanks in advance!! :)

Answer 1

does it help to know that
\[\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\]?

Answer 2

Yes! I found a sheet of a few things like that but I'm not exactly sure how they work. http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

Answer 3

so for the first one \[\cos(\theta)\tan(\theta)=\cos(\theta)\frac{\sin(\theta)}{\cos(\theta)}=\sin(\theta)\]

Answer 4

it is mostly algebra
the cosines cancel

Answer 5

They have the answer choices written out in one long line, which makes it a bit tricky to sort things out. I find it better to have it written out like this


\[\Large \cos(\theta)\tan(\theta) = \sin(\theta)\]

\[\Large \cos(\theta){\color{red}{\tan(\theta)}} = \sin(\theta)\]

\[\Large \cos(\theta){\color{red}{\frac{\sin(\theta)}{\cos(\theta)}}} = \sin(\theta)\]

At this point, I'm sure you see what cancels. Throughout the whole process, the right side stays the same.

Answer 6

the second one is completely different
it is derived from the "subtraction angle" formula
\[\sin(\alpha-\beta)=\sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)\]

Answer 7

put \(\alpha=\frac{\pi}{2}\) and \(\beta=\theta\) and you get it

Answer 8

Okay, I think I get the canceling thing. One the first one at least.
for the first question would the answer would be B since the \[\cos()\] cancel out and \[\tan( \theta)\] equals sin over cos, which equals the others. Is this correct?

Answer 9

Sorry for the weirdness, I'm not super sure how to use the equation button. I meant:
\[\cos (\theta)\]

Answer 10

yes the first one is B

Answer 11

I'm still a little stuck with the second question. Would it be C since the subtraction angle formula comes out to be
sin(π/2−θ)=sin(π2)cos(θ)−cos(π/2)sin(θ)
which matches the first part of the equation? Some of this stuff is going over my head.

Answer 12

You have it correct.

Answer 13

Thanks both of you! :3

Answer 14

you're welcome