"How to simplify sec⁡x−cos⁡xtan⁡x?" solved and explained

mooltattawsmq8

Answered question

2023-01-06

How to simplify

\frac{\sec x - \cos x}{\tan x}

Josselyncx6

Beginner2023-01-07Added 10 answers

\sec x = \frac{1}{\cos x} = \frac{\tan x}{\sin x}

So,

\frac{\sec x - \cos x}{\tan x} = \frac{\sec x}{\tan x} - \frac{\cos x}{\tan x} = \frac{\frac{\tan x}{\sin x}}{\tan x} - \frac{\cos x}{\tan x}

= \frac{1}{\sin x} - \frac{\cos x}{\tan x}

Tangent = sine/cosine, so the reciprocal of the tangent = cosine/sine

= \frac{1}{\sin x} - \frac{\cos^{2} x}{\sin x} = \frac{1 - \cos^{2} x}{\sin x}

Since

\sin^{2} x + \cos^{2} x = 1

, that means

\cos^{2} x = 1 - \sin^{2} x

= \frac{1 - (1 - \sin^{2} x)}{\sin x} = \frac{1 - 1 + \sin^{2} x}{\sin x} = \frac{\sin^{2} x}{\sin x} = \sin x

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