Question

cosx+sinxtanx simplify the equation

Answer 1

\(\bf cos(x)+sin(x){\color{brown}{ tan(x)}}\qquad cos(x)+sin(x){\color{brown}{ \cfrac{sin(x)}{cos(x)}}}
\\ \quad \\
cos(x)+\cfrac{sin^2(x)}{cos(x)}\implies \cfrac{\square ?}{cos(x)}\)

Answer 2

You can write tanx in terms of sinx and cosx. Then find the common denominator in order to add with the cosx of the right side. Taking a look at the trigonometric function will help you to simplify that expression even further.

Answer 3

if the answer is secx, what happen to the \[\sin ^2x\]

Answer 4

\[\cos(x)+\sin(x)\tan(x) = \cos(x)+\sin(x)\cfrac{\sin(x)}{\cos(x)} =
\cos(x)+\cfrac{\sin^2(x)}{\cos(x)} = \\
\cfrac{\cos^2(x)+\sin^2(x)}{\cos(x)} = \cfrac{1}{\cos(x)} = \sec(x)
\]

Answer 5

@rag4311

As you can see, the numerator becomes \[\cos^2(x) + \sin^2(x)\]just after the addition. And this is just the Pythagorean Identity, which is always equal to 1. Replace that expression by 1 and you get \[\frac{1}{\cos(x)}\]

which is the same as sec(x).