Trigonometric Identities|Please help!

Question

Please help me understand how to prove each identity. Would appreciate as much detail as possible, thanks.

a. sec x - tan x * sin x = 1 / sec x

b. 1 + cos x

⎯⎯⎯⎯ = csc x + cot x

sin x

c. sec θ tan θ

⎯⎯⎯⎯ - ⎯⎯⎯ = 1

cos θ cot θ

Answer 1

a) secx - tanxsinx = 1/secx

convert everything to sines and cosines, if you don't know what to do

Left side then = 1/cosx - sinxsinx/cosx = (1-sinxsinx)/cosx = cosxcosx/cosx = cosx

right side = 1/secx = cosx

b) left side = (1+cosx)/sinx = 1/sinx + cosx/sinx

right side = cscx + cotx = 1/sinx + cosx/sinx

c) secx/cosx - tanx/cotx = 1/cosx/cosx - sinx/cosx/cosx/sinx =

= 1/cosxcosx - sinxsinx/cosxcosx

= 1/cosxcosx - (1-cosxcosx)/cosxcosx

= 1/cosxcosx - 1/cosxcosx + cosxcosx/cosxcosx

= 0 + 1

= 1

Answer 2

Answer 3

Hi Candi! I want you to follow this secret recipe to reach the solution for this problem in the easiest possible manner!

1. sec x=1/cos x and tan x=sin x/cos x. So, substituting for both sec x and tan x, u get (1/cos x)-(sin x/cos x)(sin x)=1/sec x. This becomes (1/cos x)-((sin^2x)/(cos x))=1/sec x. Since (1) and (sin^2x) have the same denominator which is (cos x), you will combine them. It will be (1-sin^2x)/(cos x)=1/sec x. From trig identities, we know that (1-sin^2x)=cos^2x. So it will be (cos^2x)/(cos x)=1/sec x. This will simplify to (cos x)=1/sec x. Since sec x is the reciprocal of cos x, this means that the equation simplifies to (cos x)=(cos x). That's it!!! We reached there Candi!! Btw I offer my online tutoring sessions at $32/hr and I'd be happy to adjust to your schedule! Now let us continue!

1. We know that csc x=1/sin x. We also know that cot x is the reciprocal of tan x and is equal to 1/tan x which is equal to cos x/sin x. Substituting these expressions into the right side of the equation, we get: (1/sin x)+(cos x/sin x). Now, since both expressions have the same denominator when can use sin x as their common denominator and it will finally become: (1+cos x)/(sin x). We reached it!!!

1. I'm sure you know the identity that says that sec^2x-tan^2x=1. I'm sure you also know that cos x is the reciprocal of sec x and that cot x is the reciprocal of tan x. Keeping this in mind, we can take cos x to the numerator and make it sec x and now it will be multiplied by the sec x that is originally in the numerator so it will be sec^2x. Now, we will also take the cot x to the numerator so it will be converted to tan x. Since we originally have tan x in the numerator, now it will be tan^2x. So, now you have sec^2x-tan^2x=1. We got there!!! Remember, I'd be happy to adjust to your schedule!!