if `tan ((pi)/(4)+(y)/(2))=tan^(3)((pi)/(4)+(x)/(2))` then `(3+sin^(2)x)/(1+3sin^(2)x)=`

If tan((pi)/(4)+(y)/(2))=tan^(3)((pi)/(4)+(x)/(2))* Prove that (sin y)/(sin x)=(3+sin^(2)x)/(1+3sin^(2)x)

If tan((pi)/(4)+(theta)/(2))=tan^(3)((pi)/(4)+(alpha)/(2)) then sin(theta)=(sin(alpha)(3+sin^(2)(alpha)))/(1+3sin^(2)(alpha))

tan x+tan(x+(pi)/(3))+tan(x+2(pi)/(3))=3 prove that (3tan x-tan^(3)x)/(1-3tan^(2)x)=1

tan^(-1)(x/2)+tan^(-1)(x/3)=(pi)/(4)

If tan x+tan(x+(pi)/(3))+tan(x+(2 pi)/(3))=3 then prove that (3tan x-tan^(3)x)/(1-3tan^(2)x)=1

The expression (tan(x-(pi)/(2)).cos((3pi)/(2)+x)-sin^(3)((7pi)/(2)-x))/(cos(x-(pi)/(2)).tan((3pi)/(2)+x)) simplifies to

tan^(-1)2x+tan^(-1)3x=(pi)/(4)

The expression (tan(x-(pi)/(2))cos((3 pi)/(2)+x)-sin^(3)((7 pi)/(2)-x))/(cos(x-(pi)/(2))tan((3 pi)/(2)+x)) is equal to

solve: tan^(-1) ((cos x)/(1-sin x)) = ((pi)/(4)+(x)/(2)) , (-3 pi)/(2)ltxlt(pi)/(2)

The function f(x)=tan^(-1)(sin x+cos x) is an increasing function in (1)((pi)/(4),(pi)/(2))(2)(-(pi)/(2),(pi)/(4))(3)(0,(pi)/(2))(4)(-(pi)/(2),(pi)/(2))^(((pi)/(4)),(pi)/(2))^(((pi)/(4)))(2)