Verify the following identities?

Question

sin 2x=2 tan x/1+tan^2 x
sin 2θ/sin θ - cos 2θ/cos θ =sec θ
sin (x+y) cos (x-y) +cos (x+y) sin (x-y)=sin 2x
cos 2x=cot^2 x-1/cot^2 x+1

Answer 1

sin2x=2sinxcosx
2tan/1=tan^2x
=2(sinx/cosx)/sec^2x
=2sinx/cosxsec^2s
=2sinx/secx
=2sinxcosx=sin2x

2.sin2x/sinx-cos2x/cosx
2sinxcosx/sinx-(2cos^2x-1)/cosx
=2cosx-2cosx+secx
=secx

3.using the formula sinAcosB+cosAsinB=sin(A+B)
sin[(x+y)+(x-y)]
=sin2x

4.cot^2x-1/cot^2x+1
=cot^2x-1/cosec^2x
=cot^2x*sin^2x-sin^2x
=(cos^2x/sin^2x)*sin^2x-sin^2x
=cos^2x-sin^2x
=cos2x

Answer 2

sin(a+B)=sin a cos B + cos a sin B sin(a-B)=sin a cos B - cos a sin B So, sin(a+B) + sin(a-B) = sin a cos B + cos a sin B + sin a cos B - cos a sin B Which equals 2sin a cos B. So, your identity is shown.

Answer 3

I verify them!
They seem correct.

Answer 4

sin(2x) = (2tan(x))/(1 + tan(x)^2)
sin(2x) = (2(sinx/cosx))/(1 + (sinx/cosx)^2)
sin(2x) = ((2sinx)/(cosx)) / ((cos(x^2) + sin(x)^2)/(cos(x)^2)))
sin(2x) = (2sin(x)/cos(x))/(1/cos(x)^2)
sin(2x) = (2sin(x)/cos(x)) * cos(x)^2
sin(2x) = (2sin(x)cos(x)^2)/(cos(x))
sin(2x) = 2sin(x)cos(x)
sin(2x) = sin(2x)
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(sin(2x)/sin(x)) - (cos(2x)/cos(x)) = sec(x)
(2sin(x)cos(x)/sin(x)) - ((cos(x)^2 - sin(x)^2)/(cos(x))) = sec(x)
2cos(x) - ((cos(x)^2 - sin(x)^2)/(cos(x))) = sec(x)
(2cos(x)^2 - (cos(x)^2 - sin(x)^2))/(cos(x)) = sec(x)
(2cos(x)^2 - cos(x)^2 + sin(x)^2)/(cos(x)) = sec(x)
(cos(x)^2 + sin(x)^2)/(cos(x)) = sec(x)
1/(cos(x)) = sec(x)
sec(x) = sec(x)

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sin(x + y)cos(x - y) + cos(x + y)sin(x - y) = sin(2x)

Lets do it like this sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

or in our case x = (x + y) and y = (x - y), so this gives us

sin((x + y) + (x - y)) = sin(2x)
sin(x + y + x - y) = sin(2x)
sin(2x) = sin(2x)

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cos(2x) = (cot(x)^2 - 1)/(cot(x)^2 + 1)

cos(2x) = ((cos(x)/sin(x))^2 - 1)/((cos(x)/sin(x))^2 + 1)

cos(2x) = ((cos(x)^2 - sin(x)^2)/(sin(x)^2)) / ((cos(x)^2 + sin(x)^2)/(sin(x)^2))

cos(2x) = ((cos(x)^2 - sin(x)^2)/(sin(x)^2) / (1/(sin(x)^2)))
cos(2x) = ((cos(x)^2 - sin(x)^2)/(sin(x)^2) * sin(x)^2
cos(2x) = ((sin(x)^2)(cos(x)^2 - sin(x)^2))/(sin(x)^2)
cos(2x) = cos(x)^2 - sin(x)^2
cos(2x) = cos(x)^2 - sin(x)^2