1 + sin(𝜃) / cos(𝜃) + cos(𝜃) / 1 + sin(𝜃) = 2 sec(𝜃)

Question

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. (Simplify at each step.)

Answer 1

(1+sinT)/cosT + (cosT_)/(1+sinT) = 2secT

put left side over a common denominator

=(1+sinT)^2/cosT(1+sinT)+ cos^2T)/cosT(1+sinT)

=2/(cosT+cosTsinT) which equals 2secT only if you ignore cosTsinT

plug in T=60

(1+sqr3/2)/(1/2) + (1/2)/(1+sqr3/2) = left side

2(2/sqr3) =4sqr3/3 = right side

2 + sqr3 + 1/2 + sqr3/4

= 2.5 + 5sqr3/4 which doesn't = 4sqr3/3

It's not an identity as written

or

maybe you meant

(1+sinT)/(cosT+cosT)/(1+sinT) for the left side

plug in T= 180

=(1+0)/(-1-1)/(1+0)

=(1)/(-2)/(1) = -1/2

but right side 2sec180 =-2

-1/2 doesn't = -2

It's not an identity either

one counterexample disproves an identity