Answered: "Proof trigonometric identities. sec⁡(x)+tan⁡(x)=cos⁡(x)1−sin⁡(x)"

Rivka Thorpe

Answered question

2021-09-12

Proof trigonometric identities.

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

Ayesha Gomez

Skilled2021-09-13Added 104 answers

The given equation is

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

Let choose

\sec (x) + \tan (x)

to prove

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

to prove the identity.

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

(

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

)

= \frac{1 + \sin x}{\cos x}

= \frac{1 + \sin x}{\cos x} \cdot \frac{\cos x}{\cos x}

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

On further simplification,

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

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

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

= \frac{\cos x}{(1 - \sin x)}

Hence, the given identity is proved.

Jeffrey Jordon

Expert2021-12-11Added 2605 answers

Answer is given below (on video)

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