Write the trigonometric expression in terms of sine and cosine, and then simplify.
cos u + sin u tan u
To express the trigonometric expression cos u + sin u tan u in terms of sine and cosine, we can rewrite the tangent function in terms of sine and cosine.
Recall that tangent is defined as the ratio of sine to cosine:
tan u = sin u / cos u
Substituting this into the expression, we have:
cos u + sin u tan u = cos u + sin u (sin u / cos u)
Next, simplify the expression:
cos u + sin u (sin u / cos u) = cos u + sin^2 u / cos u
Now, we can rewrite sin^2 u in terms of cosine using the Pythagorean identity:
sin^2 u = 1 - cos^2 u
Substituting this into the expression, we have:
cos u + (1 - cos^2 u) / cos u
Simplify further by combining like terms:
cos u + 1 / cos u - cos^2 u / cos u
Now, simplify the last two terms:
cos u + 1 / cos u - cos u
Combine the terms:
1 / cos u
Therefore, the simplified expression is 1 / cos u, which can also be written as sec u.
cos u + sin u (sin u/cos u)
cos u + sin^2 u / cos u
cos^2u/cos u + sin^2 u/cos u
(cos^2 u + sin^2 u) / cos u
1/cos u