sect-cost=tantsint

Question

im doing trig identities and i cant prove them. please show work

Answer 1

Hi Abhi,

 

First, recall the following identities:

 

sec t = 1 / cos t

 

tan t = sin t / cos t

 

sin² t + cos² t = 1 (pythagorean identity)

         - cos² t      - cos² t

 

sin² t = 1 - cos² t (rearranged version of the pythagorean identity)

 

 

Using the first identity you can rewrite the left side of your equation as:

 

sec t - cos t = 1 / cos t - cos t

 

Then, get a common denominator and subtract:

 

1 / cos t - cos² t / cos t = (1 - cos² t) / cos t

 

Now, use the pythagorean identity replace (1 - cos² t) with sin² t:

 

(1 - cos² t) / cos t = sin² t / cos t

 

Almost home. Rewrite the above by separating out one of the sin's in sin² t:

 

sin² t / cos t = (sin t / cos t) sin t

 

Finally, rewrite (sin t / cos t) as tan t:

 

(sin t / cos t) sin t = tan t sin t

 

Bam, you're done, the left side has been transformed into the right side.

Answer 2

Hi Abhi, I'm not sure if this is the easiest way but I believe I was able to come up with a proof 

 

identity: secθ = 1/cosθ

 

(1/cosθ) - cosθ

now combine those terms with a common denominator of cos θ

(1-cos²θ) / cosθ

use identity: sin²θ + cos²θ = 1

sin²θ / cosθ

then use the fact that tanθ=sinθ/cosθ

tanθ sinθ

 

Hope this helps!