Prove-that-1-tan-x-1-tan-4x-2-

Question Number 128788 by bramlexs22 last updated on 10/Jan/21

Prove that (1 + \tan x) (1 + \tan 4 x) = 2 ?

Commented by bramlexs22 last updated on 10/Jan/21

i think its not true!

Commented by mr W last updated on 10/Jan/21

i t i s n o t t r u e!

Commented by liberty last updated on 10/Jan/21

i think it should be (1 + \tan x) (1 + \tan 44 x) = 2

Commented by mr W last updated on 10/Jan/21

t r y w i t h x = 0!

g e n e r a l l y :

(1 + \tan n x) (1 + \tan m x) \neq c o n s t a n t!

Commented by liberty last updated on 10/Jan/21

for nx + mx = \frac{π}{4} it is true!

Commented by liberty last updated on 10/Jan/21

example (1 + \tan 2 °) (1 + \tan 43 °) = 2

(1 + \tan 13 °) (1 + \tan 32 °) = 2

Commented by mr W last updated on 10/Jan/21

x i s v a r i a b l e, i t s t a n d s f o r a n y v a l u e,

n o t a p a r t i c u l a r v a l u e .

Commented by benjo_mathlover last updated on 10/Jan/21

only correct for A + B = 45 ° \Rightarrow (1 + \tan A) (1 + \tan B) = 2

Answered by Olaf last updated on 10/Jan/21

f (x) = (1 + \tan x) (1 + \tan 4 x)

f (x) = \frac{(\cos x + \sin x) (\cos 4 x + \sin 4 x)}{\cos x \cos 4 x}

f (x) = \frac{\cos x \cos 4 x + \sin x \sin 4 x + \sin x \cos 4 x + \cos x \sin 4 x}{\cos x \cos 4 x}

f (x) = \frac{\cos 3 x + \sin 5 x}{\cos x \cos 4 x}

f (x) = \frac{\cos 3 x + \cos (\frac{π}{2} - 5 x)}{\cos x \cos 4 x}

f (x) = \frac{2 \cos (\frac{3 x + \frac{π}{2} - 5 x}{2}) \cos (\frac{3 x - \frac{π}{2} + 5 x}{2})}{\cos x \cos 4 x}

f (x) = \frac{2 \cos (x - \frac{π}{4}) \cos (4 x - \frac{π}{4})}{\cos x \cos 4 x}

f (x) = 2 \Leftrightarrow \cos (x - \frac{π}{4}) \cos (4 x - \frac{π}{4}) = \cos x \cos 4 x

\Leftrightarrow \frac{1}{2} [\cos 3 x + \cos (5 x - \frac{π}{2})] = \frac{1}{2} [\cos 3 x + \cos 5 x]

\Leftrightarrow \cos (5 x - \frac{π}{2}) = \cos 5 x

\Leftrightarrow 5 x - \frac{π}{2} = \pm 5 x + 2 k π, k \in Z

\Leftrightarrow x = \frac{π}{20} + \frac{k π}{10}, k \in Z

S = {\frac{π}{20} + \frac{k π}{10}, k \in Z}

Numerically we can verify for example

that, for k = 0 :

(1 + \tan \frac{π}{20}) (1 + \tan \frac{π}{5}) = 2

Commented by liberty last updated on 10/Jan/21

correct sir

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