find-the-value-tan-2tan2-2-2-tan2-2-2-3-tan2-3-2-n-1-tan2-n-1-

Question Number 52516 by tanmay.chaudhury50@gmail.com last updated on 09/Jan/19

f i n d t h e v a l u e

t a n θ + 2 t a n 2 θ + 2^{2} t a n 2^{2} θ + 2^{3} t a n 2^{3} θ + . . + 2^{n - 1} t a n 2^{n - 1} θ

Answered by tanmay.chaudhury50@gmail.com last updated on 09/Jan/19

c o t θ - t a n θ

= \frac{1 - {t a n}^{2} θ}{t a n θ}

= 2 \times \frac{1 - {t a n}^{2} θ}{2 t a n θ} = \frac{2}{t a n 2 θ} = 2 c o t 2 θ

t a n θ = c o t θ - 2 c o t 2 θ \dots \dots \dots \dots \dots \dots \dots \dots e q n 1

2 \times t a n 2 θ = 2 \times c o t 2 θ - 2 \times 2 c o t 2^{2} θ \dots \dots e q n 2

2^{2} t a n 2^{2} θ = 2^{2} c o t 2^{2} θ - 2^{3} c o t 2^{3} θ \dots \dots \dots . . e q n 3

\dots \dots .

\dots \dots

2^{n - 1} t a n 2^{n - 1} θ = 2^{n - 1} c o t 2^{n - 1} θ - 2^{n} c o t 2^{n} θ

n o w a d d l e f t h a n d s i d e = S

w h e n a d d i n g o n l y r e d c o l o u r e d t e r m s r e m a i n

o t h e r s t e r m s c a n c e l l e d e a c h o t h e r . .

s o S = c o t θ - 2^{n} c o t 2^{n} θ

Commented by malwaan last updated on 10/Jan/19

great thank you

Commented by tanmay.chaudhury50@gmail.com last updated on 10/Jan/19

m o s t w e l c o m e \dots