Prove that sinA-1 - cosA - 1 - cosA -sinA

As we have sinA/1 + cosA = 1 cosA /sinATaking LHS sinA/1 + cosA by rationalisation.⇒sinA/1+cosA×1 cosA/1 cosA0ex0ex⇒sinA(1 cosA)/(1+cosA)(1 cosA)0ex0ex⇒sinA(1 ...

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Standard XII

Mathematics

Algebra of Derivatives

Prove that si...

Question

Prove that $\frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$

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Solution

As we have $\frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$
Taking LHS $\frac{\sin A}{1 + \cos A}$ by rationalisation.
$\Rightarrow \frac{\sin A}{1 + \cos A} \times \frac{1 - \cos A}{1 - \cos A} \Rightarrow \frac{\sin A (1 - \cos A)}{(1 + \cos A) (1 - \cos A)} \Rightarrow \frac{\sin A (1 - \cos A)}{\sin^{2} A} \Rightarrow \frac{1 - \cos A}{\sin A}$
Hence proved

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Prove that sinA1+cosA=1-cosAsinA

As we have sinA1+cosA=1-cosAsinATaking LHS sinA1+cosA by rationalisation.⇒sinA1+cosA×1-cosA1-cosA⇒sinA(1-cosA)1+cosA1-cosA⇒sinA(1-cosA)sin2A⇒1-cosAsinAHence proved

Prove that $\frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$