[tex]\huge\boxed{e) \ \alpha = 45^{o} \ \ | \ \ f) \ \alpha = 60^{o}}[/tex]
10.
e)
[tex]\frac{1-sin^{4}\alpha}{cos^{2}\alpha} = \frac{3}{2}\\\\\frac{(1-sin^{2}\alpha)(1+sin^{2}\alpha)}{cos^{2}\alpha} = \frac{3}{2}\\\\(sin^{2}\alpha + cos^{2}\alpha =1 \ \ \rightarrow \ \ cos^{2}\alpha = 1-sin^{2}\alpha), \ zatem:\\\\\frac{cos^{2}\alpha(1+sin^{2}\alpha)}{cos^{2}\alpha} =\frac{3}{2}[/tex]
[tex]1+sin^{2}\alpha = \frac{3}{2} \ \ \ |-1\\\\sin^{2}\alpha=\frac{1}{2}\\\\sin\alpha = \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \ \ \rightarrow \ \ \boxed{\alpha = 45^{o}}[/tex]
f)
[tex]\frac{sin\alpha cos^{2}\alpha + sin^{3}\alpha}{sin\alpha cos\alpha}=2\\\\\frac{sin\alpha(cos^{2}\alpha + sin^{2}\alpha)}{sin\alpha cos\alpha} = 2\\\\(sin^{2}\alpha + cos^{2}\alpha = 1), \ zatem:\\\\\frac{1}{cos\alpha} = 2 \ \ \ |\cdot cos\alpha\\\\1 = 2cos\alpha \ \ \ /:2\\\\cos\alpha = \frac{1}{2} \ \ \rightarrow \ \boxed{ \alpha = 60^{o}}[/tex]
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[tex]\huge\boxed{e) \ \alpha = 45^{o} \ \ | \ \ f) \ \alpha = 60^{o}}[/tex]
10.
e)
[tex]\frac{1-sin^{4}\alpha}{cos^{2}\alpha} = \frac{3}{2}\\\\\frac{(1-sin^{2}\alpha)(1+sin^{2}\alpha)}{cos^{2}\alpha} = \frac{3}{2}\\\\(sin^{2}\alpha + cos^{2}\alpha =1 \ \ \rightarrow \ \ cos^{2}\alpha = 1-sin^{2}\alpha), \ zatem:\\\\\frac{cos^{2}\alpha(1+sin^{2}\alpha)}{cos^{2}\alpha} =\frac{3}{2}[/tex]
[tex]1+sin^{2}\alpha = \frac{3}{2} \ \ \ |-1\\\\sin^{2}\alpha=\frac{1}{2}\\\\sin\alpha = \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \ \ \rightarrow \ \ \boxed{\alpha = 45^{o}}[/tex]
f)
[tex]\frac{sin\alpha cos^{2}\alpha + sin^{3}\alpha}{sin\alpha cos\alpha}=2\\\\\frac{sin\alpha(cos^{2}\alpha + sin^{2}\alpha)}{sin\alpha cos\alpha} = 2\\\\(sin^{2}\alpha + cos^{2}\alpha = 1), \ zatem:\\\\\frac{1}{cos\alpha} = 2 \ \ \ |\cdot cos\alpha\\\\1 = 2cos\alpha \ \ \ /:2\\\\cos\alpha = \frac{1}{2} \ \ \rightarrow \ \boxed{ \alpha = 60^{o}}[/tex]