Trigonometry

Question

I figured this out using my calculator but I need help solving through it without. 

(cosX)/(1+sinX)+tanX 

Answer 1

Calculators only give approximate answers, no matter how many decimals you get from it. They can serve as guides, but only an analytic method will get exact answers.

 

Let's use the alternative technique in:

http://www.wyzant.com/resources/answers/29169/perform_the_operation_and_simplify

 

cos(θ)/(1+sin(θ))+tan(θ) =

 

(x/r)/(1 + y/r) + y/x =

 

x/(r + y) + y/x =

 

x*x/(x*(r + y)) + y*(r + y)/(x*(r + y)) =

 

(x*x + y*(r + y))/(x*(r + y)) =

 

(x^2 + yr + y^2)/(x*(r + y)) =

 

(r^2 + yr)/(x*(r + y)) =

 

r(r + y)/(x*(r + y)) =

 

r/x = 1/cos(θ) = sec(θ).

Answer 2

cosX/(1+sinX)+tanX

 

we know that: tanX=sinX/cosX, therefore:

 

cosX/(1+sinX)+sinX/cosX

 

common denominator would be cosX(1+sinX), so the addition of these two terms would be:

 

(cosX^2+sinX+sinX^2)/cosX(1+sinX)

 

knowing sinX^2+cosX^2=1, we'll have:

 

(1+sinX)/cosX(1+sinX)

 

cancel out (1+sinX)

 

the answer would be:

 

1/cosX