Question: "How to simplify tan⁡x1+sec⁡x+1+sec⁡xtan⁡x=2csc⁡x?"

kejuuoo

Answered question

2023-01-19

How to simplify

\frac{\tan x}{1 + \sec x} + \frac{1 + \sec x}{\tan x} = 2 \csc x

sobrevenilac

Beginner2023-01-20Added 8 answers

L H S = \frac{\tan x}{1 + \sec x} + \frac{1 + \sec x}{\tan x}

= \frac{\tan x (\sec x - 1)}{(\sec x + 1) (\sec x - 1)} + \frac{1 + \sec x}{\tan x}

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

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

= \frac{\sec x - 1}{\tan x} + \frac{1 + \sec x}{\tan x}

= \frac{\sec x}{\tan x} - \frac{1}{\tan x} + \frac{1}{\tan x} + \frac{\sec x}{\tan x}

= \frac{2 \sec x}{\tan x}

= \frac{\frac{2}{\cos x}}{\frac{\sin x}{\cos x}}

= \frac{2}{\sin x}

= 2 \csc x = R H S

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