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

If tanA+sinA=m and tanA-sinA=n , then prove that m^(2)-n^(2)=4sqrt(mn) .

If "cosec" theta- sin theta=m^(3) and sec theta-cos theta=n^(3) , then prove that m^(2)n^(2)(m^(2)+n^(2))=1.

If n is a positive integer, prove that |I m(z^n)|lt=n|I m(z)||z|^(n-1.)

If m, n are positive quantities, prove that ((mn+1)/(m+1))^(m+1)gen^m .

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_(m-2,n)(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))}

If p + q = m + n and p^2 + q^2 = m^2 + n^2 , then by using mathematical induction prove that P^r + q^r = m^r + n^r .

If the ratio of the roots of the quadratic equation ax^(2)+bx+c=0 be m:n, then prove that ((m+n)^2)/(mn)=(b^(2))/(ac)

If x prop y+z , y propz+x and z prop x+y , and the constant of variation are 1,m,n respectively, then prove that, l/(l+1)+m/(m+1)+n/(n+1) = 1

If nge3 is an integer prove that 2n+1 lt 2^n by P.M.I.

If the ratio of the remainders when the polynomial (x^(2m)+1) is successively divided by (x + a) and (x + b), be b : a, prove that (a^(n)-b^(n))/(b-a)=1 , where n = 2m + 1 and m inN .