Question

Prove that :- 1+cosecA-cotA/1+cosecA+cotA=1-cosA/sinA​

Answer 1

Step-by-step explanation:

(1+cosecA-cotA)/(1+cosecA+cotA)=1-cosA/sinA

let's we take LHS,

We convert cosecA and cotA into sinA and cosA

so, (1+1/sinA-cosA/sinA)/(1+1/sinA+cosA/sinA)

after taking LCM sinA get cancelled out

so (sinA+1-cosA)/(sinA+1+cosA)

now we multiple and divide by 1-cosA

(sinA+1-cosA)/(sinA+1+cosA)×(1-cosA)/(1-cosA)

multiple both the denominators

(sinA+1-cosA)(1-cosA)/(sinA+1+cosA-sinAcosA-cosA-cos²A)

now we put 1-cos²A=sin²A,

(sinA+1-cosA)(1-cosA)/(sin²A+sinA-sinAcosA)

now take sinA common from denominator

(sinA+1-cosA)(1-cosA)/sinA(sinA+1-cosA)

now we get our answer (1-cosA)/sinA

HOPE YOU LIKE MY ANSWER

Answer 2

$\frac{1 + coseca - cota}{1 + coseca + cota} \\ \\ = \frac{ {cosec}^{2}a - {cot}^{2} a + coseca - cota }{1 + coseca + cota} \\ \\ = \frac{(coseca - cota)(coseca + cota + 1)}{1 + coseca + cota} \\ \\ = coseca - cota \\ \\ = \frac{1 - cosa}{sina}$