`cos theta-sin theta`=-1

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

If cos theta + sin theta =sqrt(2) cos theta prove that cos theta -sin theta =sqrt(2) sin theta

Prove that : sin theta cos^3 theta - cos theta sin^3 theta = 1/4 sin4theta .

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

Show that costheta [[costheta, sintheta],[-sintheta, cos theta]] + sin theta [[sintheta ,-cos theta],[costheta, sin theta]] =1

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

Prove that (sec theta+tan theta -1)/(tan theta -sec theta +1)=cos theta/(1-sin 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 ________.

cos theta + sin theta - sin 2 theta = (1)/(2), 0 lt theta lt (pi)/(2)

If theta lies in the second quadrant and tan theta=(-5)/(12) , find the value of (2cos theta)/(1-sin theta) .