How do i verify cotx cosx + sinx = cscx?

Answer 1

cotx cosx + sinx
= (cosx/sinx)cosx + sinx
= (cosx)^2/sinx + sinx
= [(cosx)^2 + (sinx)^2]/sinx
= 1/sinx >>>>(cosx)^2 + (sinx)^2 = 1
= cscx

Answer 2

difficulty a million: cscx + cotx = (a million + cosx) / sinx cscx + cotx = cscx(a million + cosx) cscx + cotx = cscx + cosxcscx cscx + cotx = cscx + cosx/sinx cscx + cotx = cscx + cotx (confirmed) difficulty 2: tan²x / secx = secx - cosx (sec²x - a million) / secx = secx - cosx cosx(sec²x - a million) = secx - cosx cosx(a million/cos²x - a million) = secx - cosx a million/cosx - cosx = secx - cosx secx - cosx = secx - cosx (confirmed) difficulty 3: cotx / cosx = cscx cosx / sinx / cosx = cscx cosx / sinxcosx = cscx a million/sinx = cscx cscx = cscx (confirmed) difficulty 4: cos²x + tan²xcos²x = a million cos²x + cos²x(sin²x / cos²x) = a million cos²x + sin²x = a million a million = a million (confirmed) desire this facilitates!

Answer 3

Turn everything into sines and cosines
use algebra and trig manipulations to turn the more complex side into the simpler side
turn the simpler side back into it original form
This method will work for most trig identities. Let's try it for this one...
Bear with me, I'm going to abuse the notation a bit and suppress the variable since all these functions are in x. This will make it a little easier to type it out. SO instead of writing cos x I'll just write cos, etc. Here goes ...
cot = cos/sin and csc = 1/sin so we have
(cos/sin) x cos + sin = 1/sin
Put together the (cos/sin) x cos, write sin as sin^2/sin, and you get
cos^2/sin + sin^2/sin = 1/sin
See the pythagoriean identity coming in? Put the left hand side together and get
(cos^2 + sin^2) / sin = 1/sin since cos^2 + cos^2 = 1
so now you have 1/sin = 1/sin = csc x. There you have it!
Try this method on other trig identities. It works nearly all the time, and soon you'll feel like a genius!

Answer 4

cotx cosx + sinx = cscx
(cos x/sinx)cosx + sin x = 1/sinx
cos^2 x/sin x +sinx = 1/ sin x
Multiply both sides by sin x, getting:
cos^2 x +sin^2 x = 1
1=1 because cos^2x + sin^2x =1

Answer 5

cotx cosx+ sinx = cscx

(use 3gonometric identty cot x = cos x / sinx)
(cosx/sinx)cos x + sinx = cscx

(then, get the gcf of the ff..which is sinx)
(cos x *cos x + sinx * sinx) / sinx = cscx

(then multiply it...)
(cos2x + sin2x )/ sin x = cscx

(ul have cos2x + sin2x = 1...8s a 3gonometric isdentity)
1/sinx = cscx

Answer 6

cot(x)cos(x) + sin(x) = csc(x)

The general method in solving these is to choose the more complex side. Once you choose that side, you convert everything to sines and cosines. From there, you try and make it equal to the other side.

In our case, we're going to choose the left hand side, which we'll denote as LHS.

LHS = cot(x)cos(x) + sin(x).

Convert the cot(x) to cos(x)/sin(x),

LHS = [cos(x)/sin(x)]cos(x) + sin(x).

Now, merge the first term into a fraction.

LHS = [cos^2(x)]/sin(x) + sin(x)

Put these two terms under a common denominator. Since the sin(x) is really divided by 1, our LCD would be sin(x), resulting in:

LHS = [cos^2(x)]/sin(x) + [sin^2(x)]/sin(x)

Merge as one fraction.

LHS = [cos^2(x) + sin^2(x)]/sin(x)

Note the numerator is now our famous identity, sin^2(x) + cos^2(x) = 1. We change it to 1.

LHS = 1/sin(x)

And now, by the definition of cosecant,

LHS = csc(x) = RHS

Answer 7

cot x cos x + sin x = csc x

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

Answer 8

1
=1^2
=|exp{i*x}|^2
=|cos(x)+i*sin(x)|^2
=sin(x)^2+cos(x)^2

Require that sin(x) is not equal to 0
divide 1 by sin(x) yields csc(x)
divide sin(x)^2+cos(x)^2 by sin(x) yields cot(x)cos(x)+sin(x)
Hence cot(x)cos(x)+sin(x)=csc(x) when sin(x) is not 0

When sin(x) is 0
cot(x)=infinity ; csc(x)=infinity.
Problem not well defined.

Answer 9

They are not verifiable because they are not equal. Set both sides equal to zero on your graphing calculator and you should see that the functions don't overlap throughout.

Answer 10

cot(x) cos(x) + sin(x) = csc(x)

cos(x)*cos(x)/sin(x) + sin(x) = 1/sin(x)

Multiply by sin(x):

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