1/sin2A+cos4A/sin4A
1/sin2x+2(cos2x)^2 -1/2sin2xcos2x
= 1/sin2x[2cos2x+2(cos2x)^2 - 1]/2cos2x
=[2cos2x+2(cos2x)^2 -1]/sin4x
To simplify the expression (1/sin^2A) + (cos^4A/sin^4A), we can start by finding a common denominator for the two terms.
The first term, 1/sin^2A, can be rewritten as sin^(-2)A. The second term, cos^4A/sin^4A, does not need any further simplification.
Now, we can combine the two terms by finding a common denominator. Since sin^(-2)A is equivalent to 1/sin^2A, the common denominator can be sin^4A.
The expression can be rewritten as (sin^(-2)A * sin^4A + cos^4A) / sin^4A.
Simplifying further, (sin^(-2)A * sin^4A) simplifies to 1, as the exponents cancel out, so we have (1 + cos^4A) / sin^4A.
To summarize, the simplified form of (1/sin^2A) + (cos^4A/sin^4A) is (1 + cos^4A) / sin^4A.
If you mean
1/sin2x + cos4x/sin4x
then that is
1/sin2x + (2cos^2 2x - 1)/(2sin2x cos2x)
= (2cos2x + 2cos^2 2x - 1)/(2sin2x cos 2x)
Not sure just what you are after