If `sinx + cosx=m` and `secx + cosecx= n` prove that `n(m^2-1)=2m`

If cos e ctheta-sintheta=m and sectheta-costheta=n , prove that (m^(2)n)^(2/3)+(m n^2)^(2/3)=1

If m= sin theta+ cos theta and n= sec theta+cosec theta , prove that n (m + 1) (m-1) = 2m .

If (sin A)/(sin B)=m , and (cosA)/(cosB)=n , prove that tan A= +- m/n sqrt ((1-n^2)/(m^2-1)) .

If m^2 + m'^2 + 2mm' costheta = 1 and n^2+n'^2+2n n'costheta=1 , (mn + m' n' + (m n' + m' n) costheta = 0 prove that m^2 + n^2 = cosec^2theta

If the direction cosines of a straight line are l,m and n, then prove that l^(2)+m^(2)+n^(2)=1

If I_(m,n)= int(sinx)^(m)(cosx)^(n) dx then prove that I_(m,n) = ((sinx)^(m+1)(cosx)^(n-1))/(m+n) +(n-1)/(m+n). I_(m,n-2)

If I_(m,n)= int(sinx)^(m)(cosx)^(n) dx then prove that I_(m,n) = ((sinx)^(m+1)(cosx)^(n-1))/(m+n) +(n-1)/(m+n). I_(m,n-2)

If m and n are two distinct numbers such that m >n, then prove that m^2-n^2 ,2mn and m^2+n^2 is a Pythagorean triplet.

If m^2+m'^2+ 2mm' cos theta =1, n^2+n'^2+2 n n' cos theta =1 and mn+m'n’ +(mn’ +m’n) cos theta=0 , prove that m^2+ n^2= cosec^2 theta .

If m=sinx+cos x+tan x+cot x+sec x+cosec x and n=sinx+cosx-tanx-cotx-secx-cosecx then (m+n-2)(m-n)=( where x is an acute angle )