if `m=3, n = 2` then prove that `m^n-n^m=1`

If in an HP, t_m=n and t_n=m then prove that t_(m+n)=(mn)/(m+n) .

if (9^(n)*3^(2)*(3^(-(n)/(2))hat -(-2))-27^(n))/(3^(3m)*2^(3))=(1)/(27) then prove that m-n=1

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

3. Ifm tan= n tan prove that 3. If m tan (0 - 3) = n tan (0+ 27). prove that cos 20 = mtmm m +n cos 20 = 2 (m - n)

If 1/ (m+i n) - (x-iy)/(x+iy) =0, where x,y,m,n are real and x+iy!=0 and m+i n!=0 , prove that m^2+n^2=1 .

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If sin alpha= (m-n)/(m+n) then prove that tan(45^@-alpha/2)=sqrt(n/m)

If S_(m)=m^(2) p and S_(n)=n^(2)p , where m ne n in an AP then prove that S_(p) =P^(3)

If A.M.and G.M.between two numbers is in the ratio m:n then prove that the numbers are in the ratio (m+sqrt(m^(2)-n^(2))):(m-sqrt(m^(2)-n^(2)))