Prove that `(1-cosalpha)/sinalpha=tan(alpha/2)`

If (2sinalpha)/(1+cosalpha +sinalpha)=y , then prove that (1-cosalpha+sinalpha )/(1+sinalpha) is also equal to y.

If (2sinalpha)/(1+cosalpha+sinalpha)=y , then prove that (1-cosalpha+sinalpha)/(1+sinalpha) is also equal to y.

If (2sinalpha)/(1+cosalpha +sinalpha)=y , then prove that (1-cosalpha+sinalpha )/(1+sinalpha) is also equal to y.

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

prove that, |{:(0,cosalpha,-sinalpha),(sinalpha,0,cosalpha),(cosalpha,sinalpha,0):}|^2=|{:(" "1,x,-x),(" "x,1," "x),(-x,x," "1):}| where x= sinalpha cosalpha

Prove that (sinalpha+cosalpha)(tanalpha+cotalpha)=secalpha+"cosec"alpha

If A = [(0,-tan(alpha//2)),(tan(alpha//2),0)] and I is a 2xx2 unit matrix, prove that I+A=(I-A)[(cosalpha,-sinalpha),(sinalpha,cosalpha)]

If A=[cosalpha+sinalpha,sqrt(2)sinalpha; -sqrt(2)sinalpha,cosalpha-sinalpha] , prove that A^n=[cosn\ alpha+sinn\ alpha, sqrt(2)sinn\ alpha; -sqrt(2)sinn\ alpha, cosn\ alpha-sinn\ alpha] for all n in Ndot

For a real number alpha , let A(alpha) denote the matrix ((cosalpha,sinalpha),(-sinalpha,cosalpha)) . Then for real numbers alpha_(1) and alpha_(2) , the value of A(alpha_(1))A(alpha_(2)) is -

If (2sinalpha)/({1+cosalpha+sinalpha})=y , then ({1-cosalpha+sinalpha})/(1+sinalpha) =