Tan x sin x + cos x= sec x

Question

I have to prove that the identity is correct. But not quite sure how to do so. 

Answer 1

note: tan(x)=sin(x)/cos(x)   and sec(x)= 1/cos(x)

then on the right side:

tan(x)sin(x) + cos(x) =[ (sin(x)) / (cos(x)) x sin(x) ] + cos(x)

[sin^2(x) + cos^2(x) / cos(x)] = 1/cos(x) =  left side = sec(x)

Answer 2

tan(x)*sin(x) + cos(x) = [sin(x) / cos(x)] sin(x) + cos(x)

 

= [sin^2(x) / cos(x)] + cos(x) 

 

= {[1 - cos^2(x)] / cos(x)} + cos(x)

 

= [1 / cos(x)] - cos(x) + cos(x)

 

= 1 / cos(x) = sec(x)