Trigonometric identity:

Applying the identities

Trigonometric identities used:

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

Applying the identities

*and***tan A = sin A / cos A***,***cot A = cos A / tan A**Simplify by removing parenthesis from the expression by the help of distribution.sin A (1 + [sin A / cos A]) + cos A (1 + [cos A / tan A])

Add together the rational expressions by lettingsin A + [sin^2 A / cos A]) + cos A + [cos^2 A / sin A])

*be the common denominator.***sin A cos A**Applying the formula(sin^2 A cos A + sin^3 A + cos^2 A sin A + cos^3 A) / (sin A cos A)

*.***a^3 + b^3 = (a + b)(a^2 - ab + b^2)**Applying the identity(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)

*,***sin^2 A + cos^2 A = 1**Factor out(sin^2 A cos A + cos^2 A sin A + (sin A + cos A)(1 - sin A cos A) ) / (sin A cos A)

*from the first two terms in the numerator,***sin A cos A**Factor out( (sin A cos A) (sin A + cos A) + (sin A + cos A)(1 - sin A cos A) ) / (sin A cos A)

*from the numerator,***(sin A + cos A)**Simplify by adding( [ sin A + cos A ] [ (sin A cos A) + (1 - sin A cos A) ] ) / (sin A cos A)

*and***sin A cos A***.***- sin A cos A**Remove unnecessary 1's and parenthesis,( [ sin A + cos A ] [ 1 ] ) / (sin A cos A)

Distribute the denominator in the numerator,(sin A + cos A) / (sin A cos A)

Cancel out the common factors from the numerator and denominator of each fraction,[ sin A / (sin A cos A) ] + [ cos A / (sin A cos A) ]

Apply the identities[ 1 / cos A ] + [ 1 / sin A ]

*and***sec A = 1 / cos A***,***cosec A = 1 / sin A**... which is the right hand side expression.sec A + cosec A

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**