" 14."(cos x)^(y)=(cos y)^(x)

If y= (cos x)^((cos x)^((cos x)……oo)) , then show that (dy)/(dx)= (y^(2) tan x)/(y.log cos x-1)

If (cos x)^y = (cos y)^x then show that (dy)/(dx) = (y tan x+log cos y)/(x tan y+log cos x).

If tan(7x^(@))=(sin x^(@)cos y^(@)+cos x^(@)sin y^(@))/(cos x^(@)cos y^(@)-sin x^(@)sin^(@)) where x+y=134^(@) ,then the least positive integral value of x, is-

cos4x-cos4y = 8 (cos x-cos y) (cos x + cos y) (cos x-sin y) (cos x + sin y)

Show that (i) cos x+cos y=2 cos (x+y)/(2) cos (x-y)/(2) (ii) cos x-cos y=-2 sin (x+y)/(2) sin (x-y)/(2) (iii) sin x+sin y=2 sin (dot(x)+y)/(2) cos (x-y)/(2) (iv) sin x-sin y=2 cos (x+y)/(2) sin (x-y)/(2)

Prove that (i) 2 cos x cos y=cos (x+y)+cos (x-y) (ii) -2 . sin x sin y=cos (x+y)-cos (x-y) (iii) 2 sin x cos y=sin (x+y)+sin (x-y) (iv) 2 cos x sin y=sin (x+y)-sin (x-y)

cos ^(-1) ""((cos x + cos y)/(1+cos x cos y))= 2 tan ^(-1) (tan ""(x)/(2) tan ""(y)/(2))

Prove that, cos 4 x - cos4 y = 8(cos x - cos y)(cos x + cos y)(cos x - sin x)(cos x + sin y)

Consider the system of equations x cos^(3) y+3x cos y sin^(2) y=14 x sin^(3) y+3x cos^(2) y sin y=13 The value of sin^(2) y+2 cos^(2) y is