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Question Number 160457 by HongKing last updated on 29/Nov/21

Simplfy:  ((1 + cos𝛂)/(sin^2 𝛂)) : (1 + (((1 + cos𝛂)/(sin𝛂)))^2 )

$$\mathrm{Simplfy}: \\ $$$$\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}}\::\:\left(\mathrm{1}\:+\:\left(\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}\boldsymbol{\alpha}}\right)^{\mathrm{2}} \right) \\ $$

Answered by MJS_new last updated on 29/Nov/21

(((1+c)/s^2 )/(1+(((1+c)/s))^2 ))=((c+1)/(s^2 +c^2 +2c+1))=(1/2)

$$\frac{\frac{\mathrm{1}+{c}}{{s}^{\mathrm{2}} }}{\mathrm{1}+\left(\frac{\mathrm{1}+{c}}{{s}}\right)^{\mathrm{2}} }=\frac{{c}+\mathrm{1}}{{s}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{c}+\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Commented by HongKing last updated on 30/Nov/21

thank you my dear Sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Sir} \\ $$

Answered by Rasheed.Sindhi last updated on 29/Nov/21

((1 + cos𝛂)/(sin^2 𝛂)) : (1 + (((1 + cos𝛂)/(sin𝛂)))^2 )  ((1 + cos𝛂)/(sin^2 𝛂)) :  ((sin^2 𝛂+1 +2cos𝛂 + cos^2 𝛂)/(sin^2 𝛂))  ((1 + cos𝛂)/(sin^2 𝛂)) :  ((2 +2cos𝛂 )/(sin^2 𝛂))  ((1 + cos𝛂)/(sin^2 𝛂)) : 2( ((1 +cos𝛂 )/(sin^2 𝛂)))  1:2

$$\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}}\::\:\left(\mathrm{1}\:+\:\left(\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}\boldsymbol{\alpha}}\right)^{\mathrm{2}} \right) \\ $$$$\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}}\::\:\:\frac{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}+\mathrm{1}\:+\mathrm{2cos}\boldsymbol{\alpha}\:+\:\mathrm{cos}^{\mathrm{2}} \boldsymbol{\alpha}}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}} \\ $$$$\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}}\::\:\:\frac{\mathrm{2}\:+\mathrm{2cos}\boldsymbol{\alpha}\:}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}} \\ $$$$\cancel{\frac{\mathrm{1}\:+\:\mathrm{cos}\boldsymbol{\alpha}}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}}}\::\:\mathrm{2}\cancel{\left(\:\frac{\mathrm{1}\:+\mathrm{cos}\boldsymbol{\alpha}\:}{\mathrm{sin}^{\mathrm{2}} \boldsymbol{\alpha}}\right)} \\ $$$$\mathrm{1}:\mathrm{2} \\ $$$$ \\ $$

Commented by HongKing last updated on 30/Nov/21

thank you my dear Sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Sir} \\ $$

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