Question

prove that sin A(1 + tanA) + cos A(1 + cot A) = secA + cosecA

Answer 1

sin A( 1+ tan A) + cos A ( 1 + cot A) = sec A +cosec A

LHS

multiplying them we get

sin A + sin A . tan A + cos A + cos A . cot A

tan A = sin A / cos A and cot A = cos A / sin A so putting the values we get

sin A + sin A . sin A / cos A + cos A + cos A . cos A / sin A

sin A + sin2A / cos A + cos A + cos2A / sin A

sin A + cos2A / sin A + cos A + sin2A / cos A

1/ sin A(sin2A + cos 2A )+ 1/ cosA ( cos2A + sin2A)

putting the value of cos2A + sin2A = 1 we get

1 / sin A + 1/ cos A

i.e. cosec A+ sec A = RHS

LHS = RHS