cos12xsin2xcos52xsin2xsin4x cosxSolutioncos12x sin2x cos52x

cos^1/2xsin^2x-cos5/2xsin^2x=sin^4x cosx

Solution

cos^1/2x sin²x - cos^5/2xsin²x = sin⁴x cosx

consider cos^1/2x sin²x - cos^5/2x sin²x

==> cos^1/2x sin²x - cos^{1/2 +2} x sin²x

==> cos^1/2x sin²x - cos^1/2x cos²x sin²x         since a^m+n = a^m aⁿ

==> cos^1/2x [ sin²x - cos²xsin²x]

==> cos^1/2x sin²x [ 1 - cos²x]

==> cos^1/2x sin²x (sin²x)     ; since sin²x + cos²x = 1==> sin²x = 1 - cos²x

==> cos^1/2x sin²⁺²x        ; since a^m aⁿ = a^m+n

==> cos^1/2x sin⁴x

==> cosx sin⁴x

Hence cos^1/2x sin²x - cos^5/2xsin²x = sin⁴x cosx