`cos x × cosx + sinx ×sinx=?`

If f(x) = \begin{vmatrix} cos2x & cos2x & sin2x \\ -cosx & cosx & -sinx\\ sinx & sinx & cosx \end{vmatrix} then

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

Find the derivative of tan^(-1)((cosx - sinx)/(cosx + sinx)) with respect to 'x'

The number of distinct real roots of the equation |{:(cosx, sinx, sinx), (sinx, cosx, sinx), (sinx, sinx, cosx):}|=0 in the interval [-\ pi/4,pi/4] is- a. 3 b. \ 4 c. \ 2 d. 1

If f(x) = [(cos x , - sinx,0),(sinx,cosx,0),(0,0,1)] then show f(x) . f(y) = f(x+y)

Find dy/dx , tan^(-1){(cosx - sinx)/(cosx + sinx)} .

Solve tan^(-1)""[(a cos x -b sinx)/(b cosx+a sinx)] , if ""(a)/(b)tanx le-1

Expand | (cosx, -sinx),(sinx , cosx)|

cosx{(cosx)/(1-sinx)+(1-sinx)/(cosx)}, is

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