1/Quadratic,Linear/Quadratic,F(x)/Quadratic /1/a sin 2x + b Sinx Cosx + Cos2x + d,1/a sinx+b Cosx + C and psinx + Qcosx + r/asinx + bcosx + C forms

1/Quadratic || Linear/Quadratic || f(x)/Quadratic || 1/(asin2x + bsinxcosx + cos2x + d) || 1/(asinx + bcosx + c) || (psinx + qcosx + r) / (asinx +bcosx + c) Forms

3 (sinx - cosx)^(4) + 6 (sinx + cosx)^(2) + 4(sin^(6)x + cos^(6)x) =

The value of 3(sinx - cosx)^4 + 6(sinx + cosx)^2 + 4 (sin^6 x + cos^6 x) is

If f(x)=(a sin x+b cosx)/(c sinx+dcosx) is decreasing for all x , then

If sinx+cosx=c , then sin^(6)x+cos^(6)x is equal to

Show that, sin3x + cos3x = (cosx-sinx)(1+4sinx cosx)

(sin^(3)x)/(1 + cosx) + (cos^(3)x)/(1 - sinx) =

Recall that sinx + cosx =u (say) and sin x cosx =v (say) are connected by (sinx +cosx)^(2) = sin^(2)x + cos^(2)x+2sin cosx rArr u^(2) = 1+2v rArr v=(u^(2)-1)/(2) It follows that any rational integral function of sinx + cosx , and sinx cosx i.e., R(sinx + cosx, sinx cosx) , or in our notation R(u,v) can be transformed to R(u, (u^(2)-1)/2) . Thus, to solve an equation of the form R(u,v)=0 , we form a polynomial equation in u and than look for solutions. The solution set of sinx + cosx -2sqrt(2) sin x cosx=0 is completely described by

Prove that (1+sinx-cosx)/(1+sinx+cosx) +(1+sinx+cosx)/(1+sinx-cosx) =2 cosec x

Prove that (1+sinx-cosx)/(1+sinx+cosx) +(1+sinx+cosx)/(1+sinx-cosx) =2 cosec x