If `P(x)=[[cos x,sin x],[-sin x,cos x]],` then show that `P(x)P(y)=P(x+y)=P(y)P(x)`

If P(x)=[(cosx, sinx),(-sinx, cosx)] , then show that P(x).P(y)=P(x+y)=P(y).P(x) .

If P(x)=[cosxsinx-sinxcosx] , then show that P(x)P(y)=P(x+y) .

If (1)/(x) , (1)/(y) , (1)/(z) are A.P. show that (y+z)/(x) , (z+x)/(y) , (x+y)/(z) are in A.P.

If P=cos(cos x)+sin (cos x) , then the least and greatest value of P respectively.

If cos(x-y), cosx and cos(x+y) are in H.P., then |cos x sec (y)/(2)| equals

For any sets X and Y prove that P(X) cup P(Y) sub P(X cup Y)

If tan x, tan y, tan z are in G.P., show that cos 2y = (cos (x +z))/(cos (x-z)).

If cos(x-y),cosx and "cos"(x+y) are in H.P., then cosxsec(y/2) is

If p(x)=sinx(sin^3x+3)+cosx(cos^3x+4)+(1/2)sin^2 2x+5, then find the range of p(x)dot

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