If `Delta_(1) = |(x,sintheta,costheta),(-sintheta,-x,1),(cos theta,1,x)|` and `Delta_(2) = |(x,sin2 theta, cos2 theta),(-sin 2theta, -x,1),(cos2 theta,1,x)|, x !=0`, then for all `theta in (0,(pi)/(2)):`

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 ________.

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 ________.

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, then for all theta in (0, (pi)/(2))

Let triangle1=|{:(x,sintheta,costheta),(-sintheta,x,1),(costheta,1,x):}| and triangle2=|{:(x,sin2theta,cos2theta),(-sin2theta,x,1),(cos2theta,1,x):}| then which of following is/are true?

,x,sin theta,cos theta-sin theta,-x,1cos theta,1,x]|=8, write the value

Prove that the determinant Delta =|{:(x,,sintheta,,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 .

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

Prove that the determinant |[x,sin theta,cos theta],[-sin theta,-x,1],[cos theta,1,x]| is independent of