Prove the following-displaystyle cosleft-frac-pi-4-xright-cosleft-frac-pi-4-yright-sinleft-frac-pi-4-xright-sinleft-frac-pi-4-yright-sin-x-y-

Question

Toppr Admin · Accepted Answer

LHS-cos-x3C0-4-x2212-x-cos-x3C0-4-x2212-y-x2212-sin-x3C0-4-x2212-x-sin-x3C0-4-x2212-y-cos-x3C0-4-x2212-x-x3C0-4-x2212-y-Using identity cosAcosB-x2212-sinAsinB-cos-A-B-cos-x3C0-2-x2212-x-y-sin-x-y-RHS-x2234-cos-x3C0-4-x2212-x-cos-x3C0-4-x2212-y-x2212-sin-x3C0-4-x2212-x-sin-x3C0-4-x2212-y-sin-x-y-Hence proved

Prove the following:cos(π4−x)cos(π4−y)−sin(π4−x)sin(π4−y)=sin(x+y)

LHS=cos(π4−x)cos(π4−y)−sin(π4−x)sin(π4−y)=cos(π4−x+π4−y)Using identity cosAcosB−sinAsinB=cos(A+B)=cos(π2−(x+y))=sin(x+y)=RHS∴cos(π4−x)cos(π4−y)−sin(π4−x)sin(π4−y)=sin(x+y)Hence proved

Prove the following:
$c o s (\frac{π}{4} - x) c o s (\frac{π}{4} - y) - s i n (\frac{π}{4} - x) s i n (\frac{π}{4} - y) = s i n (x + y)$