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.