Question

prove that 1-cosA/sinA = sinA/1+cosA​

Answer 1

Required Answer:-

Given to prove:

$\rm \mapsto \dfrac{1 - \cos(x) }{ \sin(x) } = \dfrac{ \sin(x) }{1 + \cos(x) }$

Proof:

Taking LHS,

$\rm \dfrac{1 - \cos(x) }{ \sin(x) }$

$\rm = \dfrac{(1 - \cos(x))(1 + \cos(x)) }{ \sin(x)(1 + \cos(x) )}$

$\rm = \dfrac{ {(1)}^{2} - \cos^{2} (x) }{ \sin(x)(1 + \cos(x) )}$

$\rm = \dfrac{1- \cos^{2} (x) }{ \sin(x)(1 + \cos(x) )}$

We know that,

$\rm { \sin}^{2} (x) + \cos^{2} (x) = 1$

$\implies \rm{ \sin }^{2}(x) = 1 - { \cos }^{2}(x)$

Therefore,

$\rm \dfrac{1- \cos^{2} (x) }{ \sin(x)(1 + \cos(x) )}$

$\rm = \dfrac{\sin^{2} (x) }{ \sin(x)(1 + \cos(x) )}$

$\rm = \dfrac{ \cancel{\sin(x)} \times \sin(x) }{ \cancel{\sin(x)}(1 + \cos(x) )}$

$\rm = \dfrac{\sin(x) }{1 + \cos(x) }$

= RHS

Hence, LHS = RHS (Hence Proved)

Relationship between Trigonometric Functions:

• sin(x) = 1/cosec(x)
• cos(x) = 1/sec(x)
• tan(x) = 1/cot(x)
• sin(x)/cos(x) = tan(x)

Answer 2

Answer:

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