Prove the following identity, where the angles involved (cos A − sin A + 1)/(cos A + sin A − 1) = cosec A + cot A, using the identity cosec² A = 1 + cot² A.

To prove: = cosec A + cot A We know that, ⇒ cosec 2 A = 1 + cot 2 A ⇒ 1 = cosec 2 A - cot 2 A .....(1) Dividing numerator and denominator of L.H.S. of the given equation with sin A, we get : Substituting value of 1 from equation (1) in numerator of above equation, we get : Since, L.H.S. = R.H.S. Hence, proved that = cosec A + cot A.

Mathematics

Prove the following identity, where the angles involved are acute angles for which the expressions are defined.
$\dfrac{\text{cos A - sin A + 1}}{\text{cos A + sin A - 1}}$ = cosec A + cot A, using the identity cosec² A = 1 + cot² A.

Trigonometric Identities

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Answer

To prove:

$\dfrac{\text{cos A - sin A + 1}}{\text{cos A + sin A - 1}}$ = cosec A + cot A

We know that,

⇒ cosec² A = 1 + cot² A

⇒ 1 = cosec² A - cot² A …..(1)

Dividing numerator and denominator of L.H.S. of the given equation with sin A, we get :

$\Rightarrow \dfrac{\text{cos A - sin A + 1}}{\text{cos A + sin A - 1}} \\[1em] \Rightarrow \dfrac{\dfrac{\text{cos A}}{\text{sin A}} - \dfrac{\text{sin A}}{\text{sin A}} + \dfrac{1}{\text{sin A}}}{\dfrac{\text{cos A}}{\text{sin A}} + \dfrac{\text{sin A}}{\text{sin A}} - \dfrac{1}{\text{sin A}}} \\[1em] \Rightarrow \dfrac{\text{cot A - 1 + cosec A}}{\text{cot A + 1 - cosec A}}$

Substituting value of 1 from equation (1) in numerator of above equation, we get :

$\Rightarrow \dfrac{\text{cot A + cosec A} - \text{(cosec}^2 A - \text{cot}^2 A)}{\text{cot A + 1 - cosec A}} \\[1em] \Rightarrow \dfrac{\text{cot A + cosec A} - \text{(cosec} A - \text{cot} A)\text{(cosec A + cot A)}}{\text{cot A + 1 - cosec A}} \\[1em] \Rightarrow \dfrac{\text{(cosec A + cot A)[1 - (cosec A - cot A)]}}{\text{cot A - cosec A + 1}} \\[1em] \Rightarrow \dfrac{\text{(cosec A + cot A)(cot A - cosec A + 1)}}{\text{cot A - cosec A + 1}} \\[1em] \Rightarrow \text{cosec A + cot A}.$

Since, L.H.S. = R.H.S.

Hence, proved that $\dfrac{\text{cos A - sin A + 1}}{\text{cos A + sin A - 1}}$ = cosec A + cot A.

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Mathematics

Prove the following identity, where the angles involved are acute angles for which the expressions are defined.
$\dfrac{\text{cos A - sin A + 1}}{\text{cos A + sin A - 1}}$ = cosec A + cot A, using the identity cosec² A = 1 + cot² A.

Trigonometric Identities

Answer

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