If `cosec theta + cot theta =k ,` then prove that `cos theta =(k^(2)-1)/( k^(2)+1)`

If " "a cos theta + b sin theta =c " and " b cosec theta - a sec theta = c, " prove that" cot 2 theta = (c^(2) + a^(2) - b^(2))/(2ab)

If y = cosec theta + cot theta , prove that 2(dy)/(d theta)+ y^(2)+ 1 = 0 .

Show that cot theta +tan theta =sec theta cosec theta

If k tan theta=tan k theta , then prove that (sin^(2)k theta)/(sin^(2)theta)=(k^(2))/((k^(2)-1)sin^(2)theta+1).

If p and q are the lengths of perpendiculars from the origin to the lines x cos theta - y sin theta = k cos 2 theta " and " x sec theta + y cosec theta = k , respectively, prove that p^(2) + 4q^(2) = k^(2) .

If sin theta+ sin^(2)theta=1 , then prove that cos^(2) theta+cos^(4) theta=1.

if sec theta - " cosec " theta =(4) /(3) , show that , theta =(1)/(2) sin ^(-1) ""(3)/(4) .

If (sin theta + cos theta )=sqrt(2) cos theta , show that cot theta =(sqrt(2)+1).

If " cosec"theta-sin theta=m and sec theta-cos theta=n , then prove that (m^(2)n)^((2)/(3))+(mn^(2))^((2)/(3))=1.