another trigonometric identity?

Question

. Express sec x - sin x tanx in terms of cos X


i just cant seem to get to cos x and ive tried everything i can think of. any ideas?

Answer 1

To convert the first term, just write sec x as 1/cos x

To convert the second term, note that tan(x) = sin(x)/cos(x)

So sin(x)tan(x) = sin^2(x)/cos(x)

But sin^2(x) = 1 - cos^2(x)

So sin(x)tan(x) = (1-cos^2(x))/cos(x)

Putting the two terms together gives you:

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

Answer 2

OK first off turn everything into sines and cosines. You get sec x being 1/cos x- (sin x/1)(sin x/cos x). then combine (sin x/1)(sin x/cos x) into sin^2x/cos x. You then have a common denominator and can add the two together to get 1-sin^2x/ cos x.

Using the Pythagorean identity sin^2x + cos^2x =1 you can rearrange that to be cos^2x= 1-sin^2x. Then substitute cos^2x for the 1-sin^2x in the identity and you get cos^2x/cos x. which equals cos x.

Answer 3

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

Answer 4

sec x = 1/(cos x)
tan x = sin x/cos x
sin²x + cos²x = 1

sec x - sin x tan x
= 1/(cos x) - (sin²x) / (cos x)
= (1 - sin²x) / (cos x)
= cos²x / cos x
= cos x

Answer 5

sinxtanx = csc

secx - cscx = cotx

cotx = cosx over sinx

cosx
-------- = cosx
sinx

multiply sinx by the cosx

you get....

cosx=cosxsinx

god damnit i f*cked up