Question

prove that (sec A + tan A) (1 - sin A) = cos A

Answer 1

Hello

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Given us,

( secA + tanA ) ( 1 - sinA )

we know that

secA = 1/cosA

tanA = sins/cosA

now

( 1 /cosA + sinA /cosA ) ( 1 - sinA )

= ( 1 + sinA ) ( 1 - sinA )/ cosA

= 1² - sin²A / cosA

= cos²A / cosA

= cosA

RHS

thanks

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Answer 2

Given: (sec A + tan A) (1 - sin A) = cos A

To Find: Prove L.H.S and R.H.S

Solution:

R.H.S = cos A

L.H.S

(sec A + tan A) (1 - sin A)

= sec A - sec A sin A + tan A - tan A sin A

= sec A - $\frac{sin A}{cos A}$ + tan A - $\frac{sin^{2}A }{cos A}$  [Converting sec A into $\frac{1}{cos A}$ and tan A into $\frac{sin A}{cos A}$,]

= sec A - tan A + tan A - $\frac{1-cos^{2} A}{cos A}$  [sin²A + cos²A = 1

                                                      ⇒sin²A = 1 - cos²A ]

= sec A - $\frac{1}{cos A}$ + $\frac{cos^{2}A }{cos A}$

= sec A - sec A + cos A

= cos A

∴ L.H.S = R.H.S (Proved)

Answer: L.H.S = R.H.S (Proved)