Question

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Answer 1

Answer:

above attachment is your answer.

(assume A= theta)

hope it had helped you.......

Answer 2

Question :

$\displaystyle\mathtt{1 + \frac{ {cot}^{2} \theta }{ 1 + cosec \: \theta } = cosec \: \theta}$

To prove :

LHS = RHS

Proof :

L H S

$= \displaystyle \mathtt{ 1 + \frac{ {cot}^{2} \theta }{ 1 + cosec \: \theta } }$

we know that ,

$\bigstar \mathtt{ \boxed{ \green{ {cot}^{2} \theta = {cosec}^{2} \theta - 1}}}$

Substituting the above value in the LHS we get ,

$= \displaystyle \mathtt{ 1 + \frac{ {cosec}^{2} \theta - 1 }{ 1 + cosec \: \theta } }$

we know that ,

$\bigstar \mathtt{ \boxed{ \green{ {a}^{2} - {b}^{2} = (a - b)(a + b)}}}$

hence ,

$= \displaystyle \mathtt{ 1 + \frac{ (cosec \theta - 1 )(cosec \theta + 1)}{ 1 + cosec \: \theta } }$

$= \displaystyle \mathtt{ 1 +cosec \theta - 1}$

$\mathtt{ = cosec \theta}$

As ,

$\mathtt{LHS = cosec \theta}$

$\mathtt{RHS = cosec \theta}$

hence we can conclude that ,

$\boxed{ \mathtt {\red{LHS = RHS}}}$