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Trigonometric Identities

If tan x+ xta...

Question

If (tan x +sec x)(tan y + sec y)(tanz +sec z) = (sec x - tan x )(sec y - tan y ) (secz - tan z) = K, then the value of K is

A

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B

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C

±2

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D

±1

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Solution

The correct option is D
±1

Divide and multiply
(tan x + sec x) (tan y + sec y)(tan z + sec z) to (sec x - tan x ) (sec y - tan y ) (sec z - tan z)

$\frac{(s e c x - t a n x) (s e c y - t a n y) (s e c z - t a n z) \times [(t a n x + s e c x) (t a n y + s e c y) (t a n z + s e c z)]}{(t a n x + s e c x) (t a n y + s e c y) (t a n z + s e c z)}$

$= \frac{1}{(t a n x + s e c x) (t a n y + s e c y) (t a n z + s e c z)}$

Suppose (tan x +sec x) (tan y +sec y) (tan z + sec z) =a

Then middle term becomes $\frac{1}{a}$
$a = \frac{1}{a}$
$a^{2} = 1 a = \pm 1$
So, $k = a = \pm 1$

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