Question

Solve the question in the attachment..!!

Prove that : [In the attachment] ​

Answer 1

To Prove :-

$\quad \leadsto \quad \sf \dfrac{1 + \cos A}{\sin A} + \dfrac{\sin A}{1 + \cos A} = \dfrac{2}{\sin A}$

Solution :-

Let LHS i.e ;

$\quad \leadsto \quad \sf \dfrac{1 + \cos A}{\sin A} + \dfrac{\sin A}{1 + \cos A}$

After Taking LCM can be written as ;

${ : \implies \quad \sf \dfrac{\sin A \times \sin A + ( 1 + \cos A) \cdot ( 1 + \cos A ) }{\sin A ( 1 + \cos A)}}$

Can be further written as ;

${ : \implies \quad \sf \dfrac{\sin A \times \sin A + ( 1 + \cos A)²}{\sin A ( 1 + \cos A)}}$

${ : \implies \quad \sf \dfrac{\sin²A + ( 1 + \cos A)²}{\sin A ( 1 + \cos A)}}$

${ : \implies \quad \sf \dfrac{\sin²A + 1² + (\cos A)² + 2 \times \cos A \times 1 }{\sin A(1 + \cos A)}}$

${ : \implies \quad \sf \dfrac{\sin²A + 1 + \cos²A + 2\cos A}{\sin A(1 + \cos A)}}$

can be further written as ;

${ : \implies \quad \sf \dfrac{1 + 1 + 2\cos A}{\sin A(1 + \cos A)}}$

${ : \implies \quad \sf \dfrac{2 + 2\cos A}{\sin A(1 + \cos A)}}$

${ : \implies \quad \sf \dfrac{2 ( 1 + \cos A) }{\sin A(1 + \cos A)}}$

${ : \implies \quad \sf \dfrac{2 \cancel{( 1 + \cos A)} }{\sin A \cancel{(1 + \cos A)}}}$

${ : \implies \quad \bf \dfrac{2}{\sin A}}$

$\quad \qquad { \bigstar { \underline { \boxed { \pmb { \bf { \red { \underbrace { \therefore \dfrac{1 + \cos A}{\sin A} + \dfrac{\sin A}{1 + \cos A} = \dfrac{2}{\sin A}}}}}}}}}{\bigstar}\quad \qquad$

Used Concepts :-

• $\sf ( a +b)² = a² + b² + 2ab$

• $\sf \sin² \theta + \cos² \theta = 1$

Answer 2

Step-by-step explanation:

LET US START

1+cos/sinA+sinA/1+cos A

(1+cos A)^2+sin^2 A/sin A.1+ cos A

1+cos^2 A+2cos A + sin ^2 A/ sin A. 1+ cos A

2 +2cos A/ sin (1+cos A)

2(1+cos A)/sin(1+cos A)

2/sin A

Hence

proved

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