If `e^(itheta)=costheta+isintheta,` find the value of `|1e^(ipi//3)e^(ipi//4)e^(-ipi//3)1e^(i2pi//3)e^(-ipi//4)e^(-i2pi//3)1|`

The value of int_(pi//4)^(pi//4)(e^(x)sec^(2)x)/(e^(2x)-1)dx , is

Statement-1 : If e^(itheta)=cos theta +isintheta and the value of e^(iA).e^(iB).e^(iC) is equal to -1, where A,B,C are the angles of a triangle. and Statement -2 : In any triangleABC , A + B +C = 180^(@)

A man walks a distance of 3 units from the origin towards the North-East (N45^0E) direction.From there, he walks a distance of 4 units towards the North-West (N45^0W) direction to reach a point Pdot Then, the position of P in the Argand plane is 3e^((ipi)/4)+4i (b) (3-4i)e^((ipi)/4) (4+3i)e^((ipi)/4) (d) (3+4i)e^((ipi)/4)

int_(-pi//4)^(pi//4)(e^(x)(x^(3)sinx))/(e^(2x)-1)dx equals

If A , B ,C are angles of a triangles, then the value of |[e^(2i A),e^(-i C),e^(-i B)],[e^(-i C),e^(2i B),e^(-i A)],[e^(-i B),e^(-i A),e^(2i C)]| is 1 b. -1 c. -2 d. -4

The value of lim_(x->1)(2-x)^(tan((pix)/2)) is e^(-2/pi) (b) e^(1/pi) (c) e^(2/pi) (d) e^(-1/pi)