Question

SinA(1+tanA)1cosA(1+cot)=secA+cosecA. Prove

Answer 1

Step-by-step explanation:

To Prove : sin A(1 + tan A) + cos A(1 + cot A) = sec A + cosec A

Proof :

L.H.S. = sin A(1 + tan A) + cos A(1 + cot A)

_____________________________

${\boxed{\tt{\bigstar \ \ Identity \ : \ tan A = {\dfrac{sin A}{cos A}}}}}$

${\boxed{\tt{\bigstar \ \ Identity \ : \ cot A = {\dfrac{cos A}{sin A}}}}}$

_____________________________

$\implies{\sf{ sin A + \left( 1 + {\dfrac{sin A}{cos A}} \right) + cos A \left( 1 + {\dfrac{cos A}{sin A}} \right) }}$

_____________________________

$\implies{\sf{sin A (1) + sin A \left( {\dfrac{sin A}{cos A}} \right) + cos A (1) + cos A \left( {\dfrac{cos A}{sin A}} \right) }}$

_____________________________

$\implies{\sf{sin A + {\dfrac{sin^2 A}{cos A}} + cos A + {\dfrac{cos^2 A}{sin A}} }}$

_____________________________

Rearranging the terms.

$\implies{\sf{sin A + {\dfrac{cos^2 A}{sin A}} + cos A + {\dfrac{sin^2 A}{cos A}}}}$

_____________________________

$\implies{\sf{ {\dfrac{sin A(sin A) + cos^2 A}{sin A}} + {\dfrac{cos A(cos A) + sin^2 A}{cos A}} }}$

_____________________________

$\implies{\sf{ {\dfrac{sin^2 A + cos^2 A}{sin A}} + {\dfrac{cos^2 A + sin^2 A}{cos A}}}}$

_____________________________

${\boxed{\tt{\bigstar \ \ Identity \ : \ sin^2 \theta + cos^2 \theta = 1}}}$

_____________________________

$\implies{\sf{ {\dfrac{1}{sin A}} + {\dfrac{1}{cos A}}}}$

_____________________________

${\boxed{\tt{\bigstar \ \ Identity \ : \ cosec A = {\dfrac{1}{sin A}}}}}$

${\boxed{\tt{\bigstar \ \ Identity \ : \ sec A = {\dfrac{1}{cos A}}}}}$

_____________________________

$\implies{\sf{ cosec A + sec A}}$

_____________________________

= R.H.S.

Hence, verified !!