Question

Prove/verify the identities: `(cos(-t))/(sec(-t)+tan(-t)) = 1 + sint`?

Thanks in advance

Answer 1

See below.

Explanation:

Recall that `cos(-t)=cost, sec(-t)=sect`, as cosine and secant are even functions. `tan(-t)=-tant,` as tangent is an odd function.

Thus, we have

`cost/(sect-tant)=1+sint`

Recall that `tant=sint/cost, sect=1/cost`

`cost/(1/cost-sint/cost)=1+sint`

Subtract in the denominator.

`cost/((1-sint)/cost)=1+sint`

`cost*cost/(1-sint)=1+sint`

`cos^2t/(1-sint)=1+sint`

Recall the identity

`sin^2t+cos^2t=1.` This identity also tells us that

`cos^2t=1-sin^2t`.

Apply the identity.

`(1-sin^2t)/(1-sint)=1+sint`

Using the Difference of Squares,

`(1-sin^2t)=(1+sint)(1-sint).`

`((1+sint)cancel(1-sint))/cancel(1-sint)=1+sint`

`1+sint=1+sint`

The identity holds.