Question

sinA(1+tanA)+cosA(1+cotA)= secA+cosecA​

Answer 1

To prove that:

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

Left Hand Side:

sin A (1+ tan A) + cos A(1 + cot A)

Simplifying,

= sin A + si$n^{2}$ A/ cos A + cos A + co$s^{2}$ A  /  (sin A)

= sin A + cos A + [si$n^{3}$A + co$s^{3}$ A]/ (sin A) (cos A)

=[ si$n^{2}$ A cos A + co$s^{2}$ A sin A + si$n^{3}$ A + co$s^{3}$ A]/ (sin A) (cos A)

= [ si$n^{2}$ A cos A + co$s^{3}$ A + co$s^{2}$ A sin A + si$n^{3}$ A]/ (sin A) (cos A)

= cos A (si$n^{2}$ A + co$s^{2}$ A) + sin A (si$n^{2}$ A + co$s^{2}$A)]/ (sin A) (cos A)

= cos A +sin A / (sin A) (cos A)

= (1/sin A) + (1/cos A)

= cosec A + sec A = RHS

Hence Proved

Answer 2

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

LHS = sin A(1+ tan A)+ cos A(1 + cot A)

= sin A + sin^2 A/ cos A + cos A + cos^2 A/ sin A

= sin A + cos A + [sin^3 A + cos^3 A]/sin A cos A

=[ sin^2 A cos A + cos^2 A sin A + sin^3 A + cos^3 A]/sin A cos A

= [ sin^2 A cos A +cos^3 A + cos^2 A sin A + sin^3 A]/sin A cos A

= [cos A (sin^2 A + cos^2 A) + sin A (sin^2 A + cos^2 A)]/sin A cos A

= [cos A +sin A]/sin A cos A

= (1/sin A) + (1/cos A)

= cosec A + sec A = RHS.

Proved.