Help with trig identity question A level (C3) please?

Question

I have tried to answer this trig identity question and I am completely stuck. (I would ask my teacher, but it's half term hols this week)
cot2x + cosec2x ≡ cotx
please start with:
LHS=cot2x +cosec2x
and work it from there until it gets to the stage of
=cotx
=RHS

(cotx = 1/tanx, cosecx = 1/sinx, secx = 1/cosx)
Thanks
If you use other identities (double angle formula etc.) please say which
THANK YOU =]

Answer 1

cot(2x) + csc(2x) ≡

cos(2x)/sin(2x) + 1/sin(2x) ≡

[cos(2x) + 1] / sin(2x) ≡



then i prefer that you continue this...
simply use the double angle formulas for cosine and sine...

remember that there are 3 formulas for cosine... choose the one that will remove the constant 1 on the numerator.


§

Answer 2

cot2x + cosec2x ≡ cotx

(apply identities...
cot x = cos x/sin x
cosec x = 1 / sin x)

cos2x/sin2x + 1/sin2x = cotx
(cos2x + 1)/sin2x = cotx
2 * .5 * (cos2x + 1)/sin2x = cotx

(apply cos^2 x = (cos2x + 1)/2)

2 * cos^2 x/sin2x = cotx

(apply sin2x = 2*sinx*cosx

2 * cos^2 x/ 2 * sinx * cosx = cotx

Divide top and bottom by 2 * cosx

cosx / sinx = cotx

cotx = cotx

Answer 3

cot2x+csc2x = cot x


cos2x/sin2x + 1/sin2x = cotx

with sin2x as common denominator:

(cos2x+1) / sin 2x = RHS

identities:
cos2x = 2cos^2x - 1
sin2x = 2sinxcosx

(2cos^2x -1 +1)/ ( 2 sinxcosx) = RHS
simplifying:

cosx/sinx = RHS
cot x = cot x

Answer 4

1/tan2x + 1/sin2x = cos2x/sin2x + 1/sin2x = (cos2x +1)/ sin2x

1/LHS = sin2x/(1 + cos2x) + 2sinxcosx/(1+2cossquaredx - 1)

=2sinxcosx/2cossquaredx

cancelling 1/LHS = sinx/cosx so LHS = cosx/sinx = cotx
I took A-level in 1966 so I hope I can still do this correctly.

Answer 5

A) cos 2x = (cos ^2 x - sin ^2 x) = cos^2 x ( 1- tan ^2 x) = (1- tan ^2 x)/ sec ^2 x = (1- tan ^2 x)/(1+ tan ^2 x) b) Let u=tan(π/8); u>0 tan(π/4) = 2u/(1-u^2) --> 1 =2u/(1-u^2) -->u^2+2u-1 =0 --> u= -1+√2 or u = -1 -√2. Because u>0 and so the right value of u is u =-1+√2 u^2= (-1+√2)^2 = 1-2√2 +2) = 3-2√2

Answer 6

LHS
cos 2x / sin 2x + 1 / sin 2x
[ cos 2x + 1 ] / sin 2x
2 cos ² x / sin 2x
2 cos ² x / 2 sin x cos x
cos x / sin x
cot x

RHS
cot x

LHS = RHS