Final answer:
By substituting tan x with sin x/cos x and cot x with cos x/sin x, we simplify (cos x)(tan x + sin x cot x) to sin x + cos x, proving the equation.
Step-by-step explanation:
To prove the trigonometric equation (cos x)(tanx + sinx cotx) = sinx + cos2x, we must manipulate the left-hand side to show that it is equal to the right-hand side. Begin by recalling that tan x is equal to sin x/cos x and cot x is equal to cos x/sin x. This lets us rewrite the left-hand side as follows:
cos x(tan x + sin x cot x) = cos x(sin x/cos x + sin x(cos x/sin x))
Simplify by canceling common terms:
cos x(sin x/cos x + sin x(cos x/sin x)) = cos x(sin x/cos x + 1)
This further reduces to:
sin x + cos x
Which is the right-hand side of the equation we are trying to prove. Therefore, the given trigonometric equation is proved to be true.