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

multiply the left side by sinAcosA/(sinAcosA)

sin^2A*(cosA+sinA)+cos^2A*(sin+cosA)/ ( )

(cosA+sinA)(sin^2+ cos^2)/ ( )

1/sinA + 1/Cos A

cscA + secA

qed

To prove the equation sin A(1 + tan A) + cos A(1 + cot A) = sec A + cosec A, we'll start with the left-hand side of the equation and simplify it step by step.

First, let's expand the given expression:

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

Now, we'll simplify each term using the trigonometric identities:

sin A * 1 + sin A * tan A + cos A * 1 + cos A * cot A

sin A + sin A * tan A + cos A + cos A * cot A

Next, we'll apply the definitions of tan A and cot A:

tan A = sin A / cos A
cot A = cos A / sin A

Substituting these definitions into the expression:

sin A + sin A * (sin A / cos A) + cos A + cos A * (cos A / sin A)

Now, we'll simplify further:

sin A + (sin^2 A / cos A) + cos A + (cos^2 A / sin A)

Next, let's find a common denominator for the fractions:

(sin A * sin A + cos^2 A) / cos A + (cos A * cos A + sin^2 A) / sin A

Using the Pythagorean identity (sin^2 A + cos^2 A = 1):

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

Now, we'll express the terms in terms of sec A and cosec A:

1 / cos A = sec A
1 / sin A = cosec A

Substituting these values, we get:

sec A + cosec A

Therefore, we have proved that sin A(1 + tan A) + cos A(1 + cot A) = sec A + cosec A.