Question

prove that 1+ cos A / sin A = sin A/1- cos A​

Answer 1

$\large{\mathcal{\underline{\underline{\red{SOLUTION:}}}}}\\ \\ \\{\underline{\bf To\;Prove:}}\\ \\ \\ \longrightarrow \sf \dfrac{1+\cos A}{\sin A}=\dfrac{\sin A}{1-\cos A}\\ \\ \\ {\underline{\bf By\;taking\;RHS\;part,}}\\ \\ \\ \longrightarrow \sf \dfrac{\sin A}{1-\cos A}\\ \\ \\ {\underline{\sf Now,\;rationalize\;it,}}\\ \\ \\ \longrightarrow \sf \dfrac{\sin A}{1-\cos A}\times \dfrac{1+\cos A}{1+\cos A}\\ \\ \\ \longrightarrow \sf \dfrac{\sin A(1+\cos A)}{1-\cos^{2}A}$

${\underline{\sf We\;know\;that,\;1-\cos^{2} A = \sin^{2} A}}\\ \\ \\ \longrightarrow \sf \dfrac{\sin A(1+\cos A)}{\sin^{2} A}\\ \\ \\ \longrightarrow \sf \dfrac{1+\cos A}{\sin A}\\ \\ \\ {\underline{\bf Hence\;Proved!!!}}$

Answer 2

To prove :---

• (1+cosA)/sinA = sinA/(1-cosA)

Solution :---

(1+cosA)/sinA

multiply both numinator and denominator by sinA

⟿ (1+cosA)sinA/sin²A

using (sin²A = 1-cos²A in denominator now ,,

⟿ (1+cosA)sinA/(1-cos²A)

Using (a² - b² = (a+b)(a-b) in Denominator now ,,

⟿ (1+cosA)sinA/(1+cosA)(1-cosA)

⟿ sinA/(1-cosA) = RHS

✪✪ Hence Proved ✪✪