Trigonometric identities - 13

Trigonometric identity:
sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A
The left hand side of this identity is composed of sine, cosine, tangent and cotangent functions, while the right hand side is composed of secant and cosecant functions. Since secant and cosecant functions are inverses of cosine and sine functions, hence a possible way to solve it is to rewrite the left hand side of the inequality completely in terms of sine and cosine functions.

Applying the identities tan A = sin A / cos A and cot A = cos A / tan A,
sin A (1 + [sin A / cos A]) + cos A (1 + [cos A / tan A])
Simplify by removing parenthesis from the expression by the help of distribution.
sin A + [sin^2 A / cos A]) + cos A + [cos^2 A / sin A])
Add together the rational expressions by letting sin A cos A be the common denominator.
(sin^2 A cos A + sin^3 A + cos^2 A sin A + cos^3 A) / (sin A cos A)
Applying the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2).
(sin^2 A cos A + cos^2 A sin A + (sin A + cos A)(sin^2 A - sin A cos A + cos^2 A) ) / (sin A cos A)
Applying the identity sin^2 A + cos^2 A = 1,
(sin^2 A cos A + cos^2 A sin A + (sin A + cos A)(1 - sin A cos A) ) / (sin A cos A)
Factor out sin A cos A from the first two terms in the numerator,
( (sin A cos A) (sin A + cos A) + (sin A + cos A)(1 - sin A cos A) ) / (sin A cos A)
Factor out (sin A + cos A) from the numerator,
( [ sin A + cos A ] [ (sin A cos A)  + (1 - sin A cos A) ] ) / (sin A cos A)
Simplify by adding sin A cos A and - sin A cos A.
( [ sin A + cos A ] [ 1 ] ) / (sin A cos A)
Remove unnecessary 1's and parenthesis,
(sin A + cos A) / (sin A cos A)
Distribute the denominator in the numerator,
[ sin A / (sin A cos A) ] + [ cos A / (sin A cos A) ]
Cancel out the common factors from the numerator and denominator of each fraction,
[ 1 / cos A ] + [ 1 / sin A ]
Apply the identities sec A = 1 / cos A and cosec A = 1 / sin A,
sec A + cosec A
... which is the right hand side expression.

Trigonometric identities used:
• tan A = sin A / cos A
• cot A = cos A / tan A
• sin^2 A + cos^2 A = 1
• sec A = 1 / cos A
• cosec A = 1 / sin A