\(A=cos^2x\left(1+cot^2x\right)-cot^2x+4\left(sin^2x+cos^2x\right)\)
\(=cos^2x.\left(\frac{cos^2x+sin^2x}{sin^2x}\right)-cot^2x+4\)
\(=\frac{cos^2x}{sin^2x}-cot^2x+4=4\)
\(B=2\left(cos^6x+sin^6x\right)-3\left(cos^4x+sin^4x\right)\)
\(=2\left[\left(sin^2x+cos^2x\right)^3-3sin^2x.cos^2x\left(sin^2x+cos^2x\right)\right]-3\left[\left(sin^2x+cos^2x\right)^2-2sin^2x.cos^2x\right]\)
\(=2\left(1-3sin^2x.cos^2x\right)-3\left(1-2sin^2x.cos^2x\right)\)
\(=-1-6sin^2x.cos^2x+6sin^2x.cos^2x=-1\)
\(C=\left[\left(cos^2x+sin^2x\right)^2-2sin^2x.cos^2x-1\right]\left(tan^2x+cot^2x+2tanx.cotx\right)\)
\(=-2sin^2x.cos^2x\left(tanx+cotx\right)^2\)
\(=-2\left(sinx.cosx.tanx+sinx.cosx.cotx\right)^2\)
\(=-2\left(sin^2x+cos^2x\right)^2=-2\)
