Test Index

AP EAPCET 04-Jul-2022 Shift 1 Paper

Section: Mathematics

Question : 21 of 160

Marks: +1, -0

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

n=

Solution: 👈: Video Solution

(1+\tan 1^{\circ}) (1+\tan 2^{\circ}) (1+\tan 3^{\circ}) \dots

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

Considering the following two terms

(1+\tan x) (1+\tan (45^{\circ}-x))

\;=1+\tan x+\tan (45^{\circ}-x)+\tan x \cdot \tan (45^{\circ}-x)

\;=1+ [\tan x+\tan (45^{\circ}-x)] +\tan x \cdot \tan (45^{\circ}-x)

\;=1+\tan (x+ (45^{\circ}-x) ) \cdot [1-\tan x \cdot \tan (45^{\circ}-x) ]

\; \;\; +\tan x \cdot \tan (45^{\circ}-x)

\;=1+ [1-\tan x \cdot \tan (45^{\circ}-x) ] +\tan x \cdot \tan (45^{\circ}-x)

\;=1+1=2

\;=(2)^{22} \cdot (1+\tan 45^{\circ})

\;=2^{22} \cdot 2^1=2^{23}=2^n

\; \therefore \;\; n=23

Go to Question:

Prev Question

Next Question