Verify the identity. 4 csc 2x = 2 csc²x tan x

Solution:

Given expression is

4 csc 2x = 2 csc²x tanx

To prove the identity let's solve the RHS

4 csc 2x = (2)(1/sin²x)(tanx)

= (2)(1/sin²x)(sinx/cosx)

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

= (2)(1/sinxcosx) --- (1)

The trigonometric identity used here will be;

sin2x = 2sinxcosx

sinxCosx = sin2x/2 --- (2)

Substituting (2) in (1) we get

4 csc 2x = 2/(sin2x/2)

= 4/sin2x

= 4csc2x

Hence LHS and RHS. Hence the identity stands proved.

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Verify the identity. 4 csc 2x = 2 csc²x tan x

Summary:

The identity 4 csc 2x = 2 csc²x tanx is true as per the solution above.