Question

( 1+tanA)/2sinA + (1+cotA)/2cosA=cosecA + secA ​

Answer 1

SOLUTION

TO PROVE

$\displaystyle \sf{ \bigg( \frac{1 + \tan A}{2 \sin A} \bigg) + \bigg( \frac{1 + \cot A}{2 \cos A} \bigg) = \csc A + \sec A}$

PROOF

$\displaystyle \sf{ \bigg( \frac{1 + \tan A}{2 \sin A} \bigg) + \bigg( \frac{1 + \cot A}{2 \cos A} \bigg)}$

$\displaystyle \sf{ = \bigg( \frac{1}{2 \sin A} + \frac{ \tan A}{2 \sin A} \bigg) + \bigg( \frac{1}{2 \cos A} + \frac{ \cot A}{2 \cos A} \bigg) }$

$\displaystyle \sf{ = \frac{1}{2} \bigg( \csc A + \sec A \bigg) + \frac{1}{2} \bigg( \csc A + \sec A \bigg)}$

$\displaystyle \sf{ = \bigg( \csc A + \sec A \bigg)}$

Hence proved

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Answer 2

(

2sinA

1+tanA

)+(

2cosA

1+cotA

)

\displaystyle \sf{ = \bigg( \frac{1}{2 \sin A} + \frac{ \tan A}{2 \sin A} \bigg) + \bigg( \frac{1}{2 \cos A} + \frac{ \cot A}{2 \cos A} \bigg) }=(

2sinA

1

+

2sinA

tanA

)+(

2cosA

1

+

2cosA

cotA

)

\displaystyle \sf{ = \frac{1}{2} \bigg( \csc A + \sec A \bigg) + \frac{1}{2} \bigg( \csc A + \sec A \bigg)}=

2

1

(cscA+secA)+

2

1

(cscA+secA)

\displaystyle \sf{ = \bigg( \csc A + \sec A \bigg)}=(cscA+secA)

Hence proved