Prove that: 1-cosA1+cosA = − .

We have to prove whether LHS is equal to RHS or not. =1-cosA1+cosAMultiplying LHS by 1-cosA1-cosA ⇒ = 1-cosA1+cosA × 1-cosA1-cosA ⇒ S = (1-cosA)21+cosA×1-cosA⇒ = 1-cosA1-cos2AAs 2 + 2 = 1⇒ 1 − 2 = 2Putting this in LHS, we get⇒ = 1-cosAsinA⇒ = 1sinA-cosAsinA⇒ = − ⇒ = Hence, proved.

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Prove that: $\frac{\sqrt{1 - \cos A}}{\sqrt{1 + \cos A}}$ = 𝑐𝑜𝑠𝑒𝑐 𝐴 − 𝑐𝑜𝑡𝐴.

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Detailed Solution

We have to prove whether LHS is equal to RHS or not.
𝐿𝐻𝑆 = $\frac{\sqrt{1 - \cos A}}{\sqrt{1 + \cos A}}$

Multiplying LHS by $\frac{\sqrt{1 - \cos A}}{\sqrt{1 - \cos A}}$

⇒ 𝐿𝐻𝑆 = $\frac{\sqrt{1 - \cos A}}{\sqrt{1 + \cos A}}$ × $\frac{\sqrt{1 - \cos A}}{\sqrt{1 - \cos A}}$

⇒ 𝐿𝐻S = $\frac{\sqrt{{(1 - \cos A)}^{2}}}{\sqrt{1 + \cos A \times 1 - \cos A}}$
⇒ 𝐿𝐻𝑆 = $\frac{1 - \cos A}{\sqrt{1 - \cos^{2} A}}$
As 𝑠𝑖𝑛²𝐴 + 𝑐𝑜𝑠²𝐴 = 1
⇒ 1 − 𝑐𝑜𝑠²𝐴 = 𝑠𝑖𝑛²𝐴
Putting this in LHS, we get
⇒ 𝐿𝐻𝑆 = $\frac{1 - \cos A}{\sin A}$

⇒ 𝐿𝐻𝑆 = $\frac{1}{\sin A} - \frac{\cos A}{\sin A}$
⇒ 𝐿𝐻𝑆 = 𝑐𝑜𝑠𝑒𝑐 𝐴 − 𝑐𝑜𝑡 𝐴
⇒ 𝐿𝐻𝑆 = 𝑅𝐻𝑆
Hence, proved.