given sin α = 3/5 and cos β = -4/5 , α is an acute angle and β is an obtuse angle find ;
a) cot α + cosec β
b) sin 2α tan 2β
a) cotα+1sinβ=1sin2α−1+11−cos2β=\cot{\alpha} +\frac{1}{\sin{\beta}}=\sqrt{\frac{1}{\sin^2{\alpha}}-1}+\frac{1}{\sqrt{1-\cos^2{\beta}}}=cotα+sinβ1=sin2α1−1+1−cos2β1= 259−1+11−1625=43+53=3\sqrt{\frac{25}{9}-1}+\frac{1}{\sqrt{1-\frac{16}{25}}}=\frac{4}{3}+\frac{5}{3}=3925−1+1−25161=34+35=3
b)sin2αtan2β=2sinαcosα×2sinβcosβcos2β−sin2β=\sin{2\alpha}\tan{2\beta}=2\sin{\alpha}\cos{\alpha}\times \frac{2\sin{\beta}\cos{\beta}}{\cos^2{\beta}-\sin^2{\beta}}=sin2αtan2β=2sinαcosα×cos2β−sin2β2sinβcosβ= 2sinα1−sin2α×2cosβ1−cos2βcos2β−1+cos2β=2\sin{\alpha}\sqrt{1-\sin^2{\alpha}}\times \frac{2\cos{\beta}\sqrt{1-\cos^2{\beta}}}{\cos^2{\beta}-1+\cos^2{\beta}}=2sinα1−sin2α×cos2β−1+cos2β2cosβ1−cos2β= 651−925×−851−16253225−1=2425×(−247)=−576175\frac{6}{5}\sqrt{1-\frac{9}{25}}\times \frac{-\frac{8}{5}\sqrt{1-\frac{16}{25}}}{\frac{32}{25}-1}=\frac{24}{25}\times(-\frac{24}{7})=-\frac{576}{175}561−259×2532−1−581−2516=2524×(−724)=−175576
