Show that sin 2x=2tan x/(1+tan^2x)?

Asked on by she16

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lochana2500's profile pic

lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

Posted on

L:H:S = sin2x

= 2sinx.cosx ÷ 1

Devide the numerator and denominator by cos²x

= (2sinx.cosx/cos²x) ÷ (1/cos²x) 

= 2sinx/cosx ÷ sec²x

= 2tanx ÷ (1+tan²x)

= R:H:S

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll manage the LHS.

We'll write the formula that writes the sine of double angle in terms of sine and cosine of the angle.

sin 2x = 2 sin x*cos x

Now, we'll multiply and divide by cos x the right side, to create the tangent function:

sin 2x = 2 sin x*cos x*cos x/cos x

sin 2x = 2 tan x*`cos^(2)` x

But, from Pythagorean identity, we'll have:

1 + `tan^(2)` x = 1/`cos^(2)` x => `cos^(2)` x = 1/(1+`tan^(2)` x)

sin 2x = 2 tan x/(1 + `tan^(2)` x)

Therefore, the identity sin 2x = 2 tan x/(1 + `tan^(2)` x) is verified.

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