Find the value of the expression \( \frac{\sin x + \cos x}{\cos^3 x} \). Which of the following is correct?
(A) \( 1 + \tan x + \tan^2 x - \tan^3 x \)
(B) \( 1 + \tan x + \tan^2 x + \tan^3 x \)
(C) \( 1 - \tan x + \tan^2 x + \tan^3 x \)
(D) \( (1 + \tan x) \sec^2 x \)
Answer (1)
To find the correct expression equivalent to c o s 3 x s i n x + c o s x , let us work step-by-step through the options.
We start by simplifying the original expression:
cos 3 x sin x + cos x = cos 3 x sin x + cos 3 x cos x
This can be further simplified to:
cos 3 x sin x + cos 2 x 1 = cos 3 x sin x + sec 2 x
Now convert c o s 3 x s i n x in terms of tangent:
cos 3 x sin x = cos x sin x ⋅ cos 2 x 1 = tan x ⋅ sec 2 x
Thus the expression becomes:
tan x ⋅ sec 2 x + sec 2 x = ( tan x + 1 ) sec 2 x
Therefore, the correct answer is:
(D) ( 1 + tan x ) sec 2 x
