I need help with verifying identities.

Question

anonymous · Answer

Verify the identity.

4 csc 2x = 2 csc^2x tan x

hartnn · Answer

left side = 4/sin 2x 
use 
sin 2x = 2 sin x cos x

hartnn · Answer

right side = 2 csc^2x tan x
                = 2 csc x (csc x tan x)
replace csc x = 1/ sin x, tan x = sin x/cos x

hartnn · Answer

i think it'll be easy to go from right side to left side

anonymous · Answer

so I should try to get the right side to equal the left side?

hartnn · Answer

yup.

anonymous · Answer

$$4\csc2x=\frac{ 2 }{ sinx }(\frac{ 1 }{ sinx }\frac{ sinx }{ cosx })?$$

hartnn · Answer

notice that 1 of the sin x gets cancelled.

anonymous · Answer

so $$\frac{ 1 }{ sinx }\times \frac{ sinx }{ cosx }$$

hartnn · Answer

you didn't cancel sin x from numerator...

hartnn · Answer

$\huge \frac{ 2 }{ sinx }(\frac{ 1 }{ sinx }\frac{ sinx }{ cosx })=2 \dfrac{1}{\sin x \cos x}$

anonymous · Answer

OH! okay I see what I did wrong...

hartnn · Answer

so could u get left side now ?

anonymous · Answer

what would I do now? I am not exactly sure... I was thinking of making the fraction 2cscx1/cosx?

anonymous · Answer

$$2\csc \frac{ 1 }{ \cos }$$

hartnn · Answer

$\large  \frac{ 2 }{ sinx }(\frac{ 1 }{ sinx }\frac{ sinx }{ cosx })=2 \dfrac{1}{\sin x \cos x} \ = \dfrac{4}{2\sin x \cos x}=\dfrac{4}{\sin 2x} $

anonymous · Answer

@hartnn I am pretty sure that is wrong, but I am not sure what to do otherwise.

anonymous · Answer

Oh... and then that turns into 4csc 2x