Question

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

Answer 1

\[4 \csc 2x = (2 \csc^2x) (tanx)\]

Answer 2

csc = 1/ sin

Answer 3

\[2(1/\sin(x)) = (1/\sin^2(x)) (tanx / 2)\]

Answer 4

Is that right?

Answer 5

I'm really confused.

Answer 6

So from beginning to end, what should I write?

Answer 7

ok, first left side,
4 csc 2x = 4/ sin 2x
write sin 2x = 2 sin x cos x
what u get ?

Answer 8

Should I divide both sides by 2?

Answer 9

4 / 2 sin x cos x

Answer 10

lets work with left side only
\(\large \frac{4}{sin 2x}=\frac{2}{sin x.cosx}\)
yeah, thats correct,
now right side.

Answer 11

write csc x =1/sin x
tan x = sin x/cos x
what u get ?

Answer 12

Where does the tan x go?

Answer 13

tan x was in numerator, replace it by sin x/cos x

Answer 14

2 csc^2x (sinx / cosx)

Answer 15

\(\large 2\csc^2 x . \tan x=2 \frac{1}{sin^2 x}\frac{sin x}{cos x}\)
got this ?

Answer 16

anything getting cancelled ?

Answer 17

the sinx.

Answer 18

what remains ?

Answer 19

So 2 (1/cosx)

Answer 20

\(\huge 2 \frac{1}{\sin x. \sin x}\frac{\sin x}{\cos x}=\frac{2}{\sin x\cos x }\)
got that ?

Answer 21

Yeah.

Answer 22

so, left side and right side both equal to same expression, right ?
this proves the identity

Answer 23

Oh! Wow, thanks so much!

Answer 24

your welcome ^_^