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Mathematics

The expression equivalent to sec x, is :

  1. tan⁡x+sin⁡xsec⁡x\tan x + \dfrac{\sin x}{\sec x}

  2. cos⁡x+tan⁡xcos⁡x\cos x + \dfrac{\tan x}{\cos x}

  3. cos x + tan x sin x

  4. tan x − cos x sin x

Trigonometric Identities

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Answer

Solving for option 3,

cos x + tan x sin x

⇒cos⁡x+sin⁡xcos⁡x×sin⁡x⇒cos⁡x+sin⁡2xcos⁡x⇒cos⁡2x+sin⁡2xcos⁡x⇒1cos⁡x⇒sec⁡x\Rightarrow \cos x + \dfrac{\sin x}{\cos x} \times \sin x \\[1em] \Rightarrow \cos x + \dfrac{\sin^2 x}{\cos x} \\[1em] \Rightarrow \dfrac{\cos^2 x + \sin^2 x}{\cos x} \\[1em] \Rightarrow \dfrac{1}{\cos x} \\[1em] \Rightarrow \sec x

Hence, option 3 is the correct option.

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