Question

Given sin x = 8/10, 90° <x< 180° and tan y = -12/5, 270° ≤y≤ 360°. Without finding the values for angle x and y, determine cos x cos y + sin x sin y / tan x - tan y.​​

Answer 1

Answer:
`-(189)/(208)`

Explanation:

Given

`\sin x=(8)/(10)`

x lies in the second quadrant where sin and cosec is positive and rest trigonometric functions are negative

so,
`\cos x=-√(1-sin ^2 x)\\\cos x=-\sqrt{1-((8)/(10))^2}=-\sqrt{(36)/(100)}\\\cos x=-\sqrt{(6)/(10)}`

`\tan x=(\sin x)/(\cos x)=-(4)/(3)`

similarly,
`\tan y=-(12)/(5)`

Also, y lies in 4 th Quadrant where cos and sec is positive and rest trigonometric functions are negative

`\sec y=√(1+\tan ^2y)=\sqrt{1+(-(12)/(5))^2}\\\sec y=(13)/(5)\\\cos y=(5)/(13)\\\sin y =(\tan y)/(\cos y)=-(12)/(13)`

Now putting the values

=
`(\cos x \cos y+\sin x\sin y)/(\tan x-\tan y)=(((-6)/(10))((5)/(13))+((8)/(10))((-12)/(13)))/(((-8)/(6))-((-12)/(5)))\\(\cos x \cos y+\sin x\sin y)/(\tan x-\tan y)=((-126)/(130))/((16)/(15))=-(189)/(208)`