Prove that the determinant `|(x,sin theta,cos theta),(-sin theta,-x,1),(cos theta,1,x)|` is independent of `theta`

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

(1+sin 2theta+cos 2theta)/(1+sin2 theta-cos 2 theta) =

If Delta_(1) =|{:(x, sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1,x):}| and Delta_(2) =|{:(x, sin 2theta, cos 2theta),(-sin 2theta, -x, 1),(cos 2theta, 1,x):}|, x ne 0 , then for all theta in (0, pi/2) , Then, Delta_(1) + Delta_(2) =-2x^(k) . The value of k is ________.

(1+sin2theta+cos2theta)/(1+sin2theta-cos2theta)=?

Prove that (cos^3 theta-sin^3 theta)/(cos theta-sin theta) = 1+cos theta sin theta

Prove that cos^2 theta cosec theta+sin theta=cosec theta

Prove that (cos theta+sin theta )^2 + (cos theta - sin theta )^2 =2

Prove that (sin 3 theta)/( sin theta) - (cos 3 theta)/(cos theta) = 2

Prove that : 2 sin^2 theta + 4 cos (theta + alpha) sin alpha sin theta + cos 2 (alpha + theta) is independent of theta.

Prove that ((1+cot theta +tan theta)(sin theta - cos theta))/(sec ^3 theta -cosec^3 theta) = sin^2 theta cos^2 theta.