Cosec theta = 1/sin theta
Therefore, csc theta * sin theta = 1.
cosec theta = 1/sin theta, sec theta = 1 / cos theta.
Therefore, Equation 2 reduces to,
(1/sin theta) / (1/cos theta) = cos theta/sin theta = 1/tan theta = cot theta.
tan theta + cot theta = tan theta + 1/tan theta. Taking LCM,
= (1 + tan^2 theta)/tan theta
From the identities, 1 + tan^2 theta = sec^2 theta.
Hence, equation 3 reduces to,
sec^2 theta/tan theta = (1/cos^2 theta) / (sin theta/cos theta)
= cos theta / (sin theta * cos^2 theta)
= 1 / (sin theta*cos theta). This cannot be reduced, hence is equal to sec theta*cosec theta.
1-sin^2 theta / 1+sin theta = (1 + sin theta)*(1-sin theta) / (1 + sin theta)
= 1 - sin theta.
Hope this helped :)
