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

Sagot :

xero099

Answer:

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

1) [tex]\frac{\sin A}{1 + \cos A} + \frac{\cos A}{\sin A}[/tex] Given

2) [tex]\frac{\sin^{2}A+\cos A\cdot (1+\cos A)}{\sin A\cdot (1 + \cos A)}[/tex]  [tex]\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + b\cdot c}{b\cdot d}[/tex]/Definition of power

3) [tex]\frac{\sin^{2}A+\cos^{2}A + \cos A}{\sin A\cdot (1 + \cos A)}[/tex] Distributive property/Definition of power

4) [tex]\frac{1+\cos A}{\sin A\cdot (1+\cos A)}[/tex] Associative property/[tex]\sin^{2}A + \cos^{2}A = 1[/tex]

5) [tex]\frac{1}{\sin A}[/tex] Existence of multiplicative inverse/Modulative property

6) [tex]\csc A[/tex]     [tex]\frac{1}{\sin A} = \csc A[/tex]/Result

Explanation:

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

1) [tex]\frac{\sin A}{1 + \cos A} + \frac{\cos A}{\sin A}[/tex] Given

2) [tex]\frac{\sin^{2}A+\cos A\cdot (1+\cos A)}{\sin A\cdot (1 + \cos A)}[/tex]  [tex]\frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + b\cdot c}{b\cdot d}[/tex]/Definition of power

3) [tex]\frac{\sin^{2}A+\cos^{2}A + \cos A}{\sin A\cdot (1 + \cos A)}[/tex] Distributive property/Definition of power

4) [tex]\frac{1+\cos A}{\sin A\cdot (1+\cos A)}[/tex] Associative property/[tex]\sin^{2}A + \cos^{2}A = 1[/tex]

5) [tex]\frac{1}{\sin A}[/tex] Existence of multiplicative inverse/Modulative property

6) [tex]\csc A[/tex]     [tex]\frac{1}{\sin A} = \csc A[/tex]/Result

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