If `tan frac (x)(2)= m/n`, prove than `m sin x+n cos x=n`.

If (sin beta)/(sin(2 alpha+ beta))=m/n , prove that : cot (alpha+ beta)= (n-m)/(n+m) cot alpha .

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

If x^m y^n = (x+y)^(m+n) , prove that dy/dx = y/x and (d^2y)/(dx^2) = 0

If tan x = 3/4, pi lt x lt (3 pi)/2 , find the values of sin frac (x)(2), cos frac (x)(2) and tan frac (x)(2) .

Statement -1: If 2"sin"2x - "cos" 2x=1, x ne (2n+1) (pi)/(2), n in Z, "then sin" 2x + "cos" 2x = 5 Statement-2: "sin"2x + "cos"2x = (1+2"tan" x - "tan"^(2)x)/(1+"tan"^(2)x)

If I_(m)=int (sin x+cos x)^(m)dx , then show that m l_(m)=(sin x+ cos x)^(m-1)*(sin x- cos x)+2(m-1) I_(m-2)

If f(x) = alpha x^n , prove that alpha= (f'(1))/n .

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 .

For 0 lt x lt (pi)/(2) , let P_(mn)(x)=m log_(cos x) ( sin x)+ n log_(cos x)(cotx) , where m, n in {1, 2,...,9} [For example: P_(29)(x)=2log_(cosx)(sinx)+9log_(cos x)( cot x) and " " P_(77)(x)=7 log_(cos x)(sin x)+(7 log_(cos x) ( cot x) ] On the basis of above information, answer the following questions : If P_(34)(x)=P_(22)(x) , then the value of sin x is expressed as ((sqrt(q)-1)/(p)) , then (p+q) equals

For 0 lt x lt (pi)/(2) , let P_(mn)(x)=m log_(cos x) ( sin x)+ n log_(cos x)(cotx) , where m, n in {1, 2,...,9} [For example: P_(29)(x)=2log_(cosx)(sinx)+9log_(cos x)( cot x) and " " P_(77)(x)=7 log_(cos x)(sin x)+(7 log_(cos x) ( cot x) ] On the basis of above information, answer the following questions : If P_(34)(x)=P_(22)(x) , then the value of sin x is expressed as ((sqrt(q)-1)/(p)) , then (p+q) equals