Question

Verify the identity.

4 csc 2x = 2 csc^2x tan x

Answer 1

Explanation:

work on the right hand side.

csc is the inverse of sin and tan equals sin/cos which gives us:

2(1/((sin^2)x))((sinx)/(cosx))

next cancel sinx which gives us:

2(1/(sinx))(1/(cosx))

then we multiply the numerator and denominator by 2:

4/((2sinx)(cosx))

simplify using identities:

4/2sinx

simplify using identities:

4csc2x

Answer 2

To verify the given identity, we work with the value on the right hand side. That is:

2 (csc^2 (x)) (tan x)

We know that csc is the inverse of sine and that tan = sin / tan, therefore:

= 2 (1 / sin^2 (x)) (sin x / cos x)

Cancelling sin x:

= 2 (1 / sin x) (1 / cos x)

Multiplying both the numerator and denominator by 2:

= 4 / 2 sin x cos x

From the trigonometric identities, 2 sin x cos x = sin 2x, therefore:

= 4 / sin 2x

= 4 csc 2x

Hence, it is proven.