We have an answer to "Simplify (sec⁡θ−tan⁡θ)(1+sin⁡θ)"

opatovaL

Answered question

2021-09-04

Simplify

(\sec θ - \tan θ) (1 + \sin θ)

aprovard

Skilled2021-09-05Added 94 answers

(\sec θ - \tan θ) (1 + \sin θ) =

= \sec (θ) \sin (θ) - \tan (θ) - \tan (θ) \sin (θ) =

= \frac{1}{\cos θ} + \frac{\sin θ}{\cos θ} - \tan θ - \frac{\sin θ}{\cos θ} \sin θ =

[∵ \sec θ = \frac{1}{\cos θ} and \tan θ = \frac{\sin θ}{\cos θ}]

= \frac{1}{\cos θ} + \tan θ - \tan θ - \frac{\sin^{2} θ}{\cos θ} =

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

= \frac{1 - \sin^{2} θ}{\cos θ} =

= \frac{\cos^{2} θ}{\cos θ} = \cos θ

[∵ \sin^{2} θ + \cos^{2} θ = 1]

Answer:

(\sec θ - \tan θ) (1 + \sin θ) = \cos θ

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