Prove that
`(sinalpha)/(1+cosalpha)+(1+cosalpha)/(sin alpha)=2"cosec "alpha`.

Prove that sinalpha*sin(60-alpha)sin(60+alpha) = 1/4*sin3alpha

Assertion : The system of linear equations x+(sinalpha)y+(cosalpha)z=0x+(cosalpha)y+(sinalpha)z=0x-(sinalpha)y-(cos\ alpha)z=0 has a non-trivial solution for only value of alpha lying the interval (0,\ pi) Reason: The equation in\ alpha cosalphas in alpha cosalphasinalpha cosalphasinalphac osalpha-sinalpha-cosalpha|=0 has only one solution lying in the interval (-pi/4,\ pi/4)

One side of a square makes an angle alpha with x axis and one vertex of the square is at origin. Prove that the equations of its diagonals are x(sin alpha+ cos alpha) =y (cosalpha-sinalpha) or x(cos alpha-sin alpha) + y (sin alpha + cos alpha) = a , where a is the length of the side of the square.

Givent that pi/2 lt alphaltpi then the expression sqrt((1-sinalpha)/(1+sinalpha))+sqrt((1+sinalpha)/(1-sinalpha)) (A) 1/(cosalpha) (B) - 2/(cosalpha) (C) 2/(cosalpha) (D) does not exist

A square of side a lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle alpha(0ltalphaltpi/ 4) with the positive direction of x-axis. equation its diagonal not passing through origin is (a) y(cosalpha+sinalpha)+x(sinalpha-cosalpha)="a (b) y(cosalpha+sinalpha)+x(sinalpha+cosalpha)=a (c) y(cosalpha+sinalpha)+x(cosalpha-sinalpha)=a (d) y(cosalpha-sinalpha)-x(sinalpha-cosalpha)=a

Let t_(1)=(sin alpha)^(cos alpha), t_(2)=(sin alpha)^(sin alpha), t_(3)=(cosalpha)^(cos alpha), t_(4)=(cosalpha)^(sin alpha) , where alpha in (0, (pi)/(4)) , then which of the following is correct

prove that cosalpha\ cos2alpha\ cos4alpha......cos(2^(n-1)alpha)=(sin2^nalpha)/(2^n sinalpha)\ for\ a l l\ n in N

If intsinx/(sin(x-alpha))dx=Ax+Blogsin(x-alpha)+C , then value of (A,B) is (A) (-sinalpha,cosalpha) (B) (-cosalpha,sinalpha) (C) (sinalpha,cosalpha) (D) (cosalpha,sinalpha)

Prove that: |(sinalpha, cosalpha, 1),(sinbeta, cosbeta, 1),(singamma, cosgamma, 1)|=sin(alpha-beta)+sin(beta-gamma)+sin(gamma-alpha)

If π<α<3π2 then sqrt((1-cosalpha)/(1+cosalpha))+sqrt((1+cosalpha)/(1-cosalpha)) is equal to (a) 2/(sinalpha) (b) -2/(sinalpha) (c) 1/(sinalpha) (d) -1/(sinalpha)