Question

please verify the following using double-angle identities (no shortcuts please):
\[\tan \theta = (\sin 2\theta)/(1 + \cos 2\theta)\]
\[\tan \theta = (1 - \cos 2\theta)/(\sin 2\theta)\]
\[(2\cos 2\theta)/(1 + \cos 2\theta) = 1 - \tan^{2} \theta\]

Answer 1

left side only please :)

Answer 2

left side is just the reverse of right side...if you can take the right side and make it into the left, then just reverse the work and you have it the other way...

To help you out, however, i'll do one left side :)

Answer 3

i try to put it all into sin and cos to see things better

Answer 4

Wot, no shortcuts...!

Answer 5

He's got one of them tutors....

Answer 6

i wonder what is considered a shortcut :)

Answer 7

is memorizing the formulas a shortcut?

Answer 8

Makes u do the two sides separately..duh!

Answer 9

@amistre64: sort of :)

Answer 10

lol .... um, is posting it on openstudy a shortcut?

Answer 11

i dont think so..?

Answer 12

Designer shortcut...

Answer 13

\[\tan \theta=\sin\theta/\cos\theta=(2\cos\theta\sin\theta)/(2\cos^{2}\theta)\](mulitply top and bottom by 2cos(theta)

\[=\sin(2\theta)/[1+\cos^{2}\theta-(1-\cos^{2}\theta)]=\sin(2\theta)/(1+\cos^{2}\theta-\sin^{2}\theta)=\sin(2\theta)/[1+\cos(2\theta)]\]

Answer 14

Gotta be worth a medal, that...

Answer 15

lol...would've been easier to work right to left :)

Answer 16

i know... but my teacher requires the left side... thank you though

Answer 17

someone else can do #2 and #3 ;)

Answer 18

okay.. thanks a lot! :D

Answer 19

Like i said...on 2, go from right to left then reverse your steps...it's much easier to just apply the double angle than to build it.

#3, go left to right usding the double angle formula dn it'll work out fine.

Answer 20

in fact, i solved #1 going right to left and then typed my steps in backwards. :)

Answer 21

why do you multiply 2cos theta to sin/cos?

Answer 22

to build the cos squared on the bottom. i needed to get cos^2-sin^2 on the bottom to be able to use the double-angle

Answer 23

that's the exact reason why it's often easier to work backwards...the motivation just isn't then when you have to "build" the formula instead of using it.

Answer 24

ahhh... i'm not very good at using the "multiply by 1" method.. thanks for clarifying things out for me :D

Answer 25

i think i can answer no 2 and 3 now.. thank you

Answer 26

no problem...if you start working the others and get stuck, let us know. post as far as you get and someone here can nudge you on.