Question

1+tan A/sin A +1+cot A/cos A =2 (sec A +cosec A) prove it.

Answer 1

Step-by-step explanation:

Check out the image below for the answer

Answer 2

Step-by-step explanation:

To Prove :

${\sf{{\dfrac{1 + tan A}{sin A}} + {\dfrac{1 + cot A}{cos A}} = 2 (sec A + cosec A) }}$

_______________________________

L.H.S. = ${\sf{\ \ {\dfrac{1 + tan A}{sin A}} + {\dfrac{1 + cot A}{cos A}}}}$

_______________________________

${\boxed{\sf{\bullet \ Identity \ : \ tan A = {\dfrac{sin A}{cos A}} }}}$

${\boxed{\sf{\bullet \ Identity \ : \ cot A = {\dfrac{cos A}{sin A}} }}}$

_______________________________

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

_______________________________

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

_______________________________

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

_______________________________

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

_______________________________

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

_______________________________

We can write this as :

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

_______________________________

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

_______________________________

We can write this as :

$\implies{\sf{ 2 \times {\dfrac{1}{cos A}} + 2 \times {\dfrac{1}{sin A}} }}$

_______________________________

${\boxed{\sf{\bullet \ Identity \ : \ {\dfrac{1}{cos A}} = sec A}}}$

${\boxed{\sf{\bullet \ Identity \ : \ {\dfrac{1}{sin A}} = cosec A}}}$

_______________________________

$\implies{\sf{ 2 sec A + 2 cosec A}}$

_______________________________

$\implies{\sf{2(sec A + cosec A)}}$

_______________________________

= R.H.S.

Hence, verified !!