Question

Calcule el valor de la siguiente expresión:
(sen 45° + cos 60°)(csc 45° - tan 45°)

Answer 1

Calcule el valor de la siguiente expresión:
(sen 45° + cos 60°)(csc 45° - tan 45°)
$(sen 45\° + cos 60\°)(csc 45\° - tan 45\°)= \\ \\ \\ csc 45\º= \dfrac{1}{sen\ 45\º}\qquad\qquad tan45\º = \dfrac{sen\ 45\º}{cos\ 45\º} \qquad sen\ 45\º = cos\ 45\º\\ \\ \\ (sen 45\° + cos 60\°)(\dfrac{1}{sen\ 45\º} - \dfrac{sen\ 45\º}{cos\ 45\º})=\\ \\ \\ ( \frac{ \sqrt{2}}{2} + \frac{1}{2})(\dfrac{1}{\frac{ \sqrt{2}}{2}} - \dfrac{\frac{ \sqrt{2}}{2}}{\frac{ \sqrt{2}}{2}})=$

$( \frac{ \sqrt{2}}{2} + \frac{1}{2})( \dfrac{\frac{2- \sqrt{2}}{2}}{\frac{ \sqrt{2}}{2}})= \\ \\ \\ ( \dfrac{ \sqrt{2}+1}{2} )( \dfrac{2- \sqrt{2}}{ \sqrt{2}})= \\ \\ Racionalizamos \to \dfrac{2- \sqrt{2}}{ \sqrt{2}}* \dfrac{\sqrt{2}}{ \sqrt{2}}= \dfrac{2\sqrt{2}-2}{ 2} \\ \\ \\ \left( \dfrac{ \sqrt{2}+1}{2} \right)\left( \dfrac{2\sqrt{2}-2}{ 2}\right )= \dfrac{2 \sqrt{2}^2 -2 \sqrt{2} +2 \sqrt{2}-2}{4}=$

$\dfrac{2*2 -2 \sqrt{2} +2 \sqrt{2}-2}{4}= \qquad se \ cancelan \ 2 \sqrt{2} \\ \\ \\ \dfrac{2*2 - 2}{4}= \dfrac{4-2}{4} = \dfrac{2}{4} = \boxed{ \dfrac{1}{2}} \\ \\ \\ \boxed{(sen 45\° + cos 60\°)(csc 45\° - tan 45\°)= \boxed{ \dfrac{1}{2}}}$


Espero que te sirva, salu2!!!!