Question

• If (Sec A + tan A ) (SecB + tanB) (Sec C + tanC) = (Sec A -tanA) (Sec B - tan B) (Sec C - tan C ) . Prove that each of the side is equal to ±1 .

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Answer 1

Given

(secA + tanA) (secB + tanB) (secC + tanC) = (secA - tanA) (secB - tanB) (secC - tanC)

Let

(secA + tanA) (secB + tanB) (secC + tanC) = (secA - tanA) (secB - tanB) (secC - tanC) = X

Now

➳ (secA + tanA) (secB + tanB) (secC + tanC) = (secA - tanA) (secB - tanB) (secC - tanC)

☛ Multiple '(secA - tanA) (secB - tanB) (secC - tanC)' on both sides, we get

➳ [(secA + tanA) (secB + tanB) (secC + tanC)] [(secA - tanA) (secB - tanB) (secC - tanC)] = [(secA - tanA) (secB - tanB) (secC - tanC)] [(secA - tanA) (secB - tanB) (secC - tanC)]

➳ (sec²A - tan²A) (sec²B - tan²B) (sec²C - tan²C) = (sec A - tan A)² (sec B - tan B)² (sec C - tan C)²

✰ As we know that, [sec²θ - tan²θ = 1].

➳ 1 × 1 × 1 = (sec A - tan A)² (sec B - tan B)² (sec C - tan C)²

➳ [(sec A - tan A) (sec B - tan B) (sec C - tan C)]² = 1

➳ (sec A - tan A) (sec B - tan B) (sec C - tan C) = √1

➳ (sec A - tan A) (sec B - tan B) (sec C - tan C) = ±1

➳ (sec A - tan A) (sec B - tan B) (sec C - tan C) = X = ±1 ⠀⠀⠀⠀⠀[Given]

➳ (secA + tanA) (secB + tanB) (secC + tanC) = (sec A - tan A) (sec B - tan B) (sec C - tan C) = ±1

∴ L.H.S = R.H.S = ±1⠀⠀[Hence proved]

Answer 2

${\boxed{\underline{\tt{ \orange{Required \: answer:-}}}}}$

★GIVEN:-

• $\sf{(Sec A + tan A ) (SecB + tanB) (Sec C + tanC) = (Sec A -tanA) (Sec B - tan B) (Sec C - tan C ) }</li><li>$

★TO PROVE:-

• each of the side is equal to ±1

★FORMULA USED:-

• $\sf{Sec^2x - Tan ^2x =1}$

★LET:-

• (Sec A + tan A ) (SecB + tanB) (Sec C + tanC)-----(1)

• (Sec A -tanA) (Sec B - tan B) (Sec C - tan C )--------------(2)

★SOLUTION:-

Multiply Eqs (1) and (2) we get,

$: \longrightarrow\sf{ (Sec^2A - Tan^2A)(Sec^2B - Tan ^2 B )( Sec ^2 C - Tan ^2 C ) = 1}$

$: \longrightarrow\: \sf{[(Sec A - Tan A)(Sec B - Tan B )(Sec C - Tan C )]^2 = 1}$

$: \longrightarrow\: \sf{(Sec A - Tan A)(Sec B - Tan B )(Sec C - Tan C ) = \pm \: 1}$

HENCE, each of the side is equal to ±1

EASY METHOD...