sin^2a+sin^2b-sin^2y=2sin a.sin b.cos y

[cruisebg]

sin^2a+sin^2b-sin^2y=2sin a.sin b.cos y

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[Spider Iovkov]

[tex]sin^2\alpha+sin^2\beta-sin^2\gamma=2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow sin^2\alpha+( sin\beta + sin\gamma)( sin\alpha - sin\gamma)=2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]sin^2\alpha + 2 sin{\frac{\beta+\gamma}{2}} cos{\frac{\beta-\gamma}{2}} . 2 cos{\frac{\beta+\gamma}{2}} sin{\frac{\beta-\gamma}{2}} = 2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow sin^2\alpha + sin(\beta+\gamma) sin(\beta-\gamma) = 2 sin\alpha sin\beta cos\gamma[/tex]

[tex]\alpha+\beta+\gamma=180^\circ \Leftrightarrow \beta+\gamma=180^\circ-\alpha \Rightarrow[/tex]

[tex]\Rightarrow sin^2\alpha + sin\alpha sin(\beta-\gamma) = 2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow sin\alpha [ sin\alpha + sin(\beta-\gamma) ] = 2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow sin\alpha . 2 sin{\frac{\alpha+\beta-\gamma}{2}} cos{\frac{\alpha-\beta+\gamma}{2}} 2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow sin\alpha . 2 sin{\frac{180^\circ-\gamma-\gamma}{2}} cos{\frac{180^\circ-\beta-\beta}{2}} = 2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow 2 sin\alpha sin(90^\circ-\gamma) cos(90^\circ-\beta) = 2 sin\alpha sin\beta cos\gamma \Leftrightarrow[/tex]

[tex]\Leftrightarrow 2 sin\alpha sin\beta cos\gamma = 2 sin\alpha sin\beta cos\gamma[/tex]

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