Adding (9+5cos(x)/sin(x))+(5sin(x)/1+cos(x))?

Question

How do you solve this?

(9+5cos(x)/sin(x))+(5sin(x)/1+cos(x))

(9+5cos(x)/sin(x))
+(5sin(x)/1+cos(x))

PLEASE HELP

Answer 1

You'd have to find a common denominator for the two, and then add. The common denominator is going to be (sin(x)(1 + cos(x)) Multiplying the first addend by (1 + cos(x)) gives us:

([9 + 5cos(x)]/(sin(x)) * (1 + cos(x)) / (1 + cos(x)) = [9 + 9cos(x) + 5cos(x) + 5cos^2(x)] / (sin(x)(1 + cos(x)))
= [9 + 14cos(x) + 5cos^2(x)] / (sin(x)(1 + cos(x)))

Likewise for the second addend:
[5sin(x)/(1 + cos(x)] * sin(x) / sin(x) = 5sin^2(x) / (sin(x)(1 + cos(x)))

So our numerator becomes:
[9 + 14cos(x) + 5cos^2(x)] + 5sin^2(x) = 9 + 14cos(x) + 5[cos^2(x) + sin^2(x)]
= 9 + 14cos(x) + 5
= 14 + 14cos(x)
= 14(1 + cos(x))

And our final fraction becomes:
14(1 + cos(x)) / (sin(x)(1 + cos(x))) = 14/sin(x)