Question

Please someone help me, youvmust find the value of: ​

Answer 1

1

Explanation:

Using the sum to product formulae and the exact values

sin x + sin y = 2sin(
`(x+y)/(2)` )cos(
`(x-y)/(2)` )

cos x + cos y = 2cos(
`(x+y)/(2)` )cos(
`(x-y)/(2)` )

sin45° = cos45° =
`(1)/(√(2) )` , cos30° =
`(√(3) )/(2)`

Given

`(sin75+sin15)/(cos75+cos15)`

=
`(2sin((75+15)/(2))cos((75-15)/(2)) )/(2cos((75+15)/(2))cos((75-15)/(2)) )`

=
`(2si45cos30)/(2cos45cos30)`

= (2 ×
`(1)/(√(2) )` ×
`(√(3) )/(2)` ) ÷ (2 ×
`(1)/(√(2) )` ×
`(√(3) )/(2)` )

= 1

Answer 2

Answer: 1

Explanation:

sin 75 = sin(30 + 45) = sin30 · cos45 + cos30 · sin45

`=(1)/(2)\cdot (\sqrt2)/(2)+(\sqrt3)/(2)\cdot(\sqrt2)/(2)\qquad =(\sqrt2+\sqrt6)/(4)`

sin 15 = sin(45 - 30) = sin45 · cos30 - cos45 · sin30

`=(\sqrt2)/(2)\cdot (\sqrt3)/(2)-(\sqrt2)/(2)\cdot(1)/(2)\qquad =(\sqrt6+\sqrt2)/(4)`

cos 75 = cos(30 + 45) = cos30 · cos45 - sin30 · sin45

`=(\sqrt3)/(2)\cdot (\sqrt2)/(2)-(1)/(2)\cdot(\sqrt2)/(2)\qquad =(\sqrt6-\sqrt2)/(4)`

cos 15 = cos(45 - 30) = cos45 · cos30 + sin45 · sin30

`=(\sqrt2)/(2)\cdot (\sqrt3)/(2)+(\sqrt2)/(2)\cdot(1)/(2)\qquad =(\sqrt6+\sqrt2)/(4)`

`(\sin 75+\sin 15)/(\cos 75+\cos 15)`

`=((\sqrt2+\sqrt6)/(4)+(\sqrt6-\sqrt2)/(4))/((\sqrt6-\sqrt2)/(4)+(\sqrt6+\sqrt2)/(4))\\\\\\=((\sqrt6)/(2))/((\sqrt6)/(2))\\\\\\=1`