Prove that the determinant
`Delta =|{:(x,,sin0,,cos 0),(-sin 0,,-x,,1),(cos 0,,1,,x):}|`
is independent of 0 .

Let f(x) = |{:(cos x ,1,0 ),(1,2cosx,1),(0,1,2cosx):}| then

The value of the determinants |{:(1,a,a^(2)),(cos(n-1)x,cos nx , cos(n+1)x),(sin(n-1)x , sin nx , sin(n+1)x):}| is zero if

Show that the determinant Delta (x) is given by Delta (x) = |{:(sin(x+alpha),cos(x+alpha),a+xsinalpha),(sin(x+beta),cos(x+beta),b+xsinbeta),(sin(x+gamma),cos(x+gamma),c+xsingamma):}| is independent of x.

Statement I The system of linear equations x+(sin alpha )y+(cos alpha )z=0 x+(cos alpha ) y+(sin alpha )z=0 -x+(sin alpha )y-(cos alpha )z=0 has a not trivial solution for only one value of alpha lying between 0 and pi . Statement II |{:(sin x, cos x , cos x),( cos x , sin x , cos x) , (cos x , cos x , sin x ):}|=0 has no solution in the interval -pi//4 lt x lt pi//4 .

Prove that: sin 12^0 sin 48^0 sin 54^0=1/8

Solve the following integration int (sin x+cos x)/(sqrt(1+sin 2x))*dx , here (sin x + cos x) gt 0

Show that the function f given by f(x)=tan^(-1)(sin x + cos x), x gt 0 is always an increasing function in (0, (pi)/(4)) .

Prove that f(x)= {((sin x)/(x)+ cos x ",",x ne 0),(2",",x=0):} is a continuous function at x= 0

cos(2pi -x) cos (-x) - sin(2pi +x) sin (-x) = 0