Verify the following identities cos3 theta 1 sin theta 1 s

Verify the following identities:

cos³ theta / 1+ sin theta = 1- sin theta / sec theta

Hint: Write cos³ theta = cos theta cos² theta

and use pythagorean identity followed by a factorization. Lastly, recall cos theta= 1/ sec theta

Solution

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

As cos^3 theta = cos theta* cos^2 theta

RHS : cos^3 theta / (1+ sin theta) = cos theta* cos^2 theta /(1+ sin theta)

= costheta( 1- sin^2theta) / (1+ sin theta)

= costheta( 1 + sintheta)(1-sintheta)/( 1+sintheta)

= costheta( 1- sintheta)

= costheta - sintheta*costheta

Now costheta = 1/sectheta

So, = costheta - sintheta*costheta

= costheta - sintheta/sectheta

= RHS