Question

prove that Cos A * cot A /1+sin A = cosec A-1

Answer 1

Given :  CosA * cotA/ ( 1 + SinA)  = Cosec A - 1

To Find : Prove

Solution:

CosA * cotA/ ( 1 + SinA)  = Cosec A - 1

LHS = CosA * cotA/ ( 1 + SinA)

= Cos A. Cos A/(SinA )(1 + SinA)

= Cos²A/ SinA ( 1 + SinA)

= ( 1 - Sin²A) /SinA(1 + SinA)

= (1 + SinA)(1 - SinA)/(SinA.(1 + SinA))

=( 1  - SinA)/SinA

= 1/SinA  - SinA/SinA

= CosecA - 1

= RHS

QED

Hence proved

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Answer 2

Given :- prove that Cos A * cot A /1+sin A = cosec A-1 ?

Solution :-

→ Cos A * cot A /1+sin A = cosec A-1

putting cot A = cos A/sin A in LHS,

→ cos²A / sin A(1 + sin A) = cosec A -1

putting cos²A = 1 - sin²A in LHS ,

→ (1 - sin²A) / sin A(1 + sin A) = cosec A -1

using a² - b² = (a + b)(a - b)

→ (1 - sinA)(1 + sinA) / sin A(1 + sin A) = cosec A -1

→ (1 - sinA) / sin A = cosec A -1

→ 1/ sinA - sinA / sinA = cosec A -1

→ cosec A -1 = cosec A -1 (Proved)

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