Question

(cosecA)/(cot A + tan A) = cos A​

Answer 1

Answer:

Step-by-step explanation:

$\bf \mathbb{TO \: PROVE :}$

$\dfrac{cosecA}{cotA+tanA}=cosA$

$\textit{How \:To \:Do :}$

First we need to take L.H.S and we need to convert 'cotA and tanA' in terms of 'sinA and cosA' and we need to take L.C.M of tem after simplifying by applying the value of a trigonometric identity and by converting 'cosecA' in terms of 'sinA' we can prove that L.H.S = R.H.S

$\mathfrak{FORMULA \, \: REQUIRED}:$

1 ) cotA = cosA/sinA

2)  tanA = sinA/cosA

3) sin²A + cos²A = 1

4) cosecA = 1/sinA

$\bf\cal{SOLUTION}:$

Taking L.H.S :-

$=\dfrac{cosecA}{cotA+tanA}$

$=\dfrac{cosecA}{\dfrac{cosA}{sinA}+\dfrac{sinA}{cosA}}$

[ ∴ cotA = cosA/sinA , tanA = sinA/cosA]

Taking L.C.M :-

$=\dfrac{cosecA}{\dfrac{cosA(cosA)+sinA(sinA)}{sinA\times cosA}}$

$=\dfrac{cosecA}{\dfrac{cos^2A+sin^2A}{sinA cosA}}$

$=\dfrac{cosecA}{\dfrac{1}{sinA cosA}}$

[ ∴ sin²A + cos²A = cos²A + sin²A = 1]

$=cosecA\times sinA cosA$

$=\dfrac{1}{sinA}\times sinA cosA$

[ ∴ cosecA = 1/sinA]

Cancelling the common terms :-

= cosA

= R.H.S

Hence Proved that '(cosecA)/(cot A + tan A) = cos A​'.