verify the following identities using a trigonometric proof.

Question

a) csctheta/costheta-cottheta=tantheta

 

 

 

b)1+csctheta/sectheta=costheta+cottheta

Answer 1

Hi Paula,

 

Please use parentheses to specify terms!!

For a, I am assuming, the expression on the left is (csctheta/costheta)-cottheta.

csctheta = 1/sintheta

csctheta/costheta = ((1/sintheta)/costheta) = 1/(sinthetacostheta)

cottheta = costheta/sintheta

(csctheta/costheta)-cottheta = 

    1                           costheta

----------------------  -  -----------   multiply the second expression by costheta

sinthetacostheta        sintheta

 

    1 - cos^2theta               sin^2theta                sintheta

= ---------------------   =  ----------------------  = --------------  = tantheta

   sinthetacostheta             sinthetacostheta         costheta

 

b) I will assume that the expression is (1+csctheta)/sectheta

csctheta = 1/sintheta and sectheta = 1/costheta

1+csctheta = 1 + 1

------------- multiply 1 by sintheta

sintheta

sintheta + 1 1 (sintheta + 1)costheta sinthetacostheta + costheta

=------------------- / ---------------- = ---------------------------------- = ---------------------------------------------

sintheta costheta sintheta sintheta

costheta

= ------------- + costheta = costheta + cottheta

sintheta