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Question Number 91038 by  M±th+et+s last updated on 27/Apr/20
if sin((α/2))=(4/5)  and cos((β/2))=(3/5)  prove  sin(α)=cos(β)
$${if}\:{sin}\left(\frac{\alpha}{\mathrm{2}}\right)=\frac{\mathrm{4}}{\mathrm{5}} \\ $$$${and}\:{cos}\left(\frac{\beta}{\mathrm{2}}\right)=\frac{\mathrm{3}}{\mathrm{5}} \\ $$$${prove} \\ $$$${sin}\left(\alpha\right)={cos}\left(\beta\right) \\ $$
Commented by jagoll last updated on 27/Apr/20
sin α = 2sin ((α/2))cos ((α/2))  = 2×(4/5)×(3/5) = ((24)/(25))  cos β = 2sin ((β/2))cos ((β/2))  = 2×(3/5)×(4/5)= ((24)/(25))
$$\mathrm{sin}\:\alpha\:=\:\mathrm{2sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right) \\ $$$$=\:\mathrm{2}×\frac{\mathrm{4}}{\mathrm{5}}×\frac{\mathrm{3}}{\mathrm{5}}\:=\:\frac{\mathrm{24}}{\mathrm{25}} \\ $$$$\mathrm{cos}\:\beta\:=\:\mathrm{2sin}\:\left(\frac{\beta}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{\beta}{\mathrm{2}}\right) \\ $$$$=\:\mathrm{2}×\frac{\mathrm{3}}{\mathrm{5}}×\frac{\mathrm{4}}{\mathrm{5}}=\:\frac{\mathrm{24}}{\mathrm{25}} \\ $$
Answered by $@ty@m123 last updated on 27/Apr/20
sin^2 (α/2)+cos^2 (β/2)=((4^2 +3^2 )/5^2 )  sin^2 (α/2)+cos^2 (β/2)=1  ⇒(α/2)=(β/2)  ⇒α=β
$$\mathrm{sin}^{\mathrm{2}} \frac{\alpha}{\mathrm{2}}+\mathrm{cos}\:^{\mathrm{2}} \frac{\beta}{\mathrm{2}}=\frac{\mathrm{4}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} }{\mathrm{5}^{\mathrm{2}} } \\ $$$$\mathrm{sin}^{\mathrm{2}} \frac{\alpha}{\mathrm{2}}+\mathrm{cos}\:^{\mathrm{2}} \frac{\beta}{\mathrm{2}}=\mathrm{1} \\ $$$$\Rightarrow\frac{\alpha}{\mathrm{2}}=\frac{\beta}{\mathrm{2}} \\ $$$$\Rightarrow\alpha=\beta \\ $$

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