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If-cos-3-5-0-lt-lt-2-Find-tan-2-45-2-3-




Question Number 206425 by hardmath last updated on 13/Apr/24
If   cos𝛂 = (3/5)   (0<𝛂<(𝛑/2))  Find:   ((tan^2  (45° + (𝛂/2)))/3) = ?
$$\mathrm{If}\:\:\:\mathrm{cos}\boldsymbol{\alpha}\:=\:\frac{\mathrm{3}}{\mathrm{5}}\:\:\:\left(\mathrm{0}<\boldsymbol{\alpha}<\frac{\boldsymbol{\pi}}{\mathrm{2}}\right) \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{tan}^{\mathrm{2}} \:\left(\mathrm{45}°\:+\:\frac{\boldsymbol{\alpha}}{\mathrm{2}}\right)}{\mathrm{3}}\:=\:? \\ $$
Answered by MM42 last updated on 13/Apr/24
tan^2 a=((1−cos2a)/(1+cos2a))  ⇒tan^2 (45+(a/2))=((1−cos(90+a))/(1+cos(90+a)))  =((1+sina)/(1−sina))=((1+(4/5))/(1−(4/5)))=9⇒ans=3 ✓
$${tan}^{\mathrm{2}} {a}=\frac{\mathrm{1}−{cos}\mathrm{2}{a}}{\mathrm{1}+{cos}\mathrm{2}{a}} \\ $$$$\Rightarrow{tan}^{\mathrm{2}} \left(\mathrm{45}+\frac{{a}}{\mathrm{2}}\right)=\frac{\mathrm{1}−{cos}\left(\mathrm{90}+{a}\right)}{\mathrm{1}+{cos}\left(\mathrm{90}+{a}\right)} \\ $$$$=\frac{\mathrm{1}+{sina}}{\mathrm{1}−{sina}}=\frac{\mathrm{1}+\frac{\mathrm{4}}{\mathrm{5}}}{\mathrm{1}−\frac{\mathrm{4}}{\mathrm{5}}}=\mathrm{9}\Rightarrow{ans}=\mathrm{3}\:\checkmark \\ $$$$ \\ $$

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