Question

sin 35° cos 55° + cos 35º sin 55°/
cosec2 10° - tan2 80°​

Answer 1

Question :

$\sf \: \dfrac{ \sin(35^{\circ}) \cos(55^{\circ}) + \cos(35^{\circ}) \sin(55 ^{\circ})}{ { \csc(10^{\circ}) }^{2} - \tan(80^{\circ})^{2} }$

To find :

Value of given trigonometric ratio

Identity used :

$\sf \bullet \: \sin(90^{\circ} - x) \: = \: \cos(x) \\ \sf \bullet \: \cos(90^{\circ} - x) \: = \sin(x) \\ \sf \bullet \: \tan(90^{\circ} - x) \: = \cot(x) \\ \sf \bullet \: ({ \sin(x) })^{2} + ( { \cos(x) })^{2} = 1 \\ \sf \bullet \: ( { \csc(x) })^{2} - ({ \cot(x) })^{2} = 1$

Solution :

$\sf \: \dfrac{ \sin(35^{\circ}) \cos(55^{\circ}) + \cos(35^{\circ}) \sin(55 ^{\circ})}{ { (\csc(10^{\circ}) })^{2} - (\tan(80^{\circ}))^{2} } \\ \\ \\ = \sf \: \dfrac{ \sin(35^{\circ}) \cos(90^{\circ} - 35^{\circ}) + \cos(35^{\circ}) \sin(90^{\circ} -35 ^{\circ})}{ { (\csc(10^{\circ}) })^{2} - ( \tan( 90^{\circ} - 10^{\circ}))^{2} } \\ \\ \\ = \sf \: \dfrac{ \sin(35^{\circ}) \times \sin(35^{\circ}) + \cos(35^{\circ}) \times \cos(35^{\circ})}{ { (\csc(10^{\circ}) })^{2} - ( \cot( 10^{\circ}))^{2} } \: \\ \\ \\ = \sf \: \dfrac{( \sin(35^{\circ}) )^{2} + (\cos(35^{\circ}))^{2})}{1 } \\ \\ \\ = \sf \: \dfrac{1}{1} \\ \\ = \bold{1}$

ANSWER : 1