How do you prove $(1+cosx)/sinx + sinx/(1+cosx) = 4$ ?

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Answer 1 · 2016-02-23T14:04:29

$(1+cosx)/sinx+sinx/(1+cosx)=2cscx$

(not $4$ as stated in the question)

$(1+cosx)/sinx+sinx/(1+cosx)$

= $((1+cosx)^2+sin^2x)/(sinx(1+cosx)$

= $(1+cos^2x+2cosx+sin^2x)/(sinx(1+cosx)$

= $(1+2cosx+sin^2x+cos^2x)/(sinx(1+cosx)$

= $(2+2cosx)/(sinx(1+cosx)$ as $sin^2x+cos^2x=1$

= $(2(1+cosx))/(sinx(1+cosx)$

= $2/sinx$ = $2cscx$

Answer 2 · 2016-02-23T15:37:16

The equation has solutions x = 2 arc tan ( 2 $+-$ $sqrt$ 3)

Converting to functions of x/2, the equation reduces to a quadratic in tan x/2. The roots are tan x/2 = 2 $+-$ $sqrt$ 3

How do you prove (1+cosx)/sinx + sinx/(1+cosx) = 4(1+cosx)/sinx + sinx/(1+cosx) = 4?