Question

write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression. sec beta - cos beta.

A.1-cos^2beta
B.sin^2beta
C.cos^2beta
D.1
E.sin beta tan beta

Answer 1

Answer:

E

Step-by-step explanation:

The expression given is:  sec β - cos β ( i will write "beta" instead of the symbol β)

We know that $sec(x)=\frac{1}{cos(x)}$

Substituting this into the the expression and doing some algebra manipulation, we have:

$sec(beta)-cos(beta)\\=\frac{1}{cos(beta)}-cos(beta)\\=\frac{1-cos^{2}(beta)}{cos(beta)}$

Using the identity  $cos^{2}(x)+sin^{2}(x)=1$ (also $sin^{2}(x)=1-cos^{2}(x)$), we can now write:

$\frac{1-cos^{2}(beta)}{cos(beta)}\\=\frac{sin^2(beta)}{cos(beta)}\\=\frac{sin(beta)*sin(beta)}{cos(beta)}\\=\frac{sin(beta)}{cos(beta)}*sin(beta)$

We know $tan(x)=\frac{sin(x)}{cos(x)}$, substituting this, we have:

$\frac{sin(beta)}{cos(beta)}*sin(beta)\\=tan(beta)*sin(beta)$

Answer choice E is the right answer.