Question

sec A (1-sinA) (secA+tanA)= ???​

Answer 1

Step-by-step explanation:

Given :

$\rightarrowtail \bold{secA(1-sinA)(secA + tan A) }$

To prove :

$\rightarrowtail \bold{secA(1-sinA)(secA + tan A)=1 }$

Solution :

$\mapsto \bold{secA(1-sinA)(secA + tan A) }$

• ∵ secA = 1/cosA
• ∵ tanA = sinA/cosA

$\mapsto \bold{\dfrac{1}{cosA} (1-sinA)(\dfrac{1}{cosA})(\dfrac{sinA}{cosA}) }$

$\mapsto \bold{(\dfrac{1-sinA}{cosA})(\dfrac{1+sinA}{cosA}) }$

• ∴ Taking L.C.M

$\mapsto \bold{\dfrac{(1-sinA)(1+sinA)}{(cosA)^{2}} }$

• ∵ (a + b)(a - b) = a² - b²

$\mapsto \bold{\dfrac{(1)^{2}-(sin^{2}A)}{cos^{2}A} }$

• ∵ sin²A + cos²A = 1
• ∴ 1 - sin²A = cos²A

$\mapsto \bold{\dfrac{cos^{2}A}{cos^{2}A} }$

$\dashrightarrow \bold{1}$

• ∴ L.H.S. = R.H.S