Question

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

Answer 1

We proceed to demonstrate the identity given on statement by algebraic and trigonometric means:

1)
`(\sin A)/(1 + \cos A) + (\cos A)/(\sin A)` Given

2)
`(\sin^(2)A+\cos A\cdot (1+\cos A))/(\sin A\cdot (1 + \cos A))`
`(a)/(b) + (c)/(d) = (a\cdot d + b\cdot c)/(b\cdot d)`/Definition of power

3)
`(\sin^(2)A+\cos^(2)A + \cos A)/(\sin A\cdot (1 + \cos A))` Distributive property/Definition of power

4)
`(1+\cos A)/(\sin A\cdot (1+\cos A))` Associative property/
`\sin^(2)A + \cos^(2)A = 1`

5)
`(1)/(\sin A)` Existence of multiplicative inverse/Modulative property

6)
`\csc A`
`(1)/(\sin A) = \csc A`/Result

Step-by-step explanation:

We proceed to demonstrate the identity given on statement by algebraic and trigonometric means:

1)
`(\sin A)/(1 + \cos A) + (\cos A)/(\sin A)` Given

2)
`(\sin^(2)A+\cos A\cdot (1+\cos A))/(\sin A\cdot (1 + \cos A))`
`(a)/(b) + (c)/(d) = (a\cdot d + b\cdot c)/(b\cdot d)`/Definition of power

3)
`(\sin^(2)A+\cos^(2)A + \cos A)/(\sin A\cdot (1 + \cos A))` Distributive property/Definition of power

4)
`(1+\cos A)/(\sin A\cdot (1+\cos A))` Associative property/
`\sin^(2)A + \cos^(2)A = 1`

5)
`(1)/(\sin A)` Existence of multiplicative inverse/Modulative property

6)
`\csc A`
`(1)/(\sin A) = \csc A`/Result