If 1-tan 1-1-tan 2- - - -1-tan 45-2n- then n isA- 22B- 24C- 23D- 12

The correct option is C 23tan(45^∘ x+x)=1 ⇒tan(45^∘ x)+tan x1 tan(45^∘ x)tan x=1 ⇒ (1+tan x)(1+tan(45^∘ x))=2 (1+tan 1^∘)(1+tan44^∘)=2 (1+tan 2^∘)(1+tan43^∘) ...

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Byju's Answer

Standard XII

Mathematics

Principal Solution of Trigonometric Equation

If 1+tan 1∘1+...

Question

If $(1 + tan 1^{\circ}) (1 + tan 2^{\circ}) . . . . . . . . (1 + tan 45^{\circ}) = 2^{n},$ then $n$ is

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Solution

The correct option is C $23$
$tan (45^{\circ} - x + x) = 1 \Rightarrow \frac{tan (45^{\circ} - x) + tan x}{1 - tan (45^{\circ} - x) tan x} = 1 \Rightarrow (1 + tan x) (1 + tan (45^{\circ} - x)) = 2$

$(1 + tan 1^{\circ}) (1 + tan 44^{\circ}) = 2 (1 + tan 2^{\circ}) (1 + tan 43^{\circ}) = 2 (1 + tan 3^{\circ}) (1 + tan 42^{\circ}) = 2 . . . (1 + tan 22^{\circ}) (1 + tan 23^{\circ}) = 2$

$\Rightarrow (1 + tan 1^{\circ}) (1 + tan 2^{\circ}) . . . . . . . . (1 + tan 45^{\circ}) = 2^{23}$

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Similar questions

Q. If

(1 + tan 1^{\circ}) \cdot (1 + tan 2^{\circ}) \cdot (1 + tan 3^{\circ}) \dots \cdot (1 + tan 45^{\circ}) = 2^{n}

, then

^{'} n^{'}

is equal to ?

Q. Given that

(1 + tan 1^{\circ}) (1 + tan 2^{\circ}) . . . . (1 + tan 45^{\circ}) = 2^{\circ},

find n.

Q. The value of

A = (1 + tan 1^{\circ}) (1 + tan 2^{\circ}) (1 + tan 3^{\circ}) \dots (1 + tan 45^{\circ}) is

Q. The value of

(1 + tan 1^{\circ}) (1 + tan 2^{\circ}) \dots (1 + tan 45^{\circ})

Q. If

L = (4 tan 1^{\circ}) (4 tan 2^{\circ}) (4 tan 3^{\circ}) \dots (4 tan 10^{\circ}),

M = (tan 2^{\circ}) (tan 4^{\circ}) (tan 6^{\circ}) \dots (tan 20^{\circ}),

and

N = (1 + tan 1^{\circ}) (1 - tan 1^{\circ}) (1 + tan 2^{\circ}) (1 - tan 2^{\circ}) \dots (1 + tan 10^{\circ}) (1 - tan 10^{\circ})

, then the value of

\frac{L}{128 M N}

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If (1+tan1∘)(1+tan2∘)........(1+tan45∘)=2n, then n is

The correct option is C 23tan(45∘−x+x)=1⇒tan(45∘−x)+tanx1−tan(45∘−x)tanx=1⇒(1+tanx)(1+tan(45∘−x))=2 (1+tan1∘)(1+tan44∘)=2(1+tan2∘)(1+tan43∘)=2(1+tan3∘)(1+tan42∘)=2...(1+tan22∘)(1+tan23∘)=2 ⇒(1+tan1∘)(1+tan2∘)........(1+tan45∘)=223

If $(1 + tan 1^{\circ}) (1 + tan 2^{\circ}) . . . . . . . . (1 + tan 45^{\circ}) = 2^{n},$ then $n$ is