## anonymous one year ago Verify the identity. cosine of x divided by quantity one plus sine of x plus quantity one plus sine of x divided by cosine of x equals two times secant of x.

1. misty1212

HI!!

2. misty1212

lets see if we can write this in math ok?

3. misty1212

$\frac{\cos(x)}{1+\sin(x)} +\frac{1+\sin(x)}{\cos(x)}$ right ?

4. anonymous

$\frac{ cosx }{ 1+sinx }+\frac{ 1+sinx }{ cosx }= 2\sec x$

5. misty1212

yeah must be , since the answer is $$2\sec(x)$$ lets do the addition and see it

6. anonymous

Isn't it 2sec x

7. misty1212

yes it is

8. anonymous

because everything else cancels out. yay

9. misty1212

lets do the addition and see it trick i learned via @satellite73 put $$\cos(x)=a$$, and $$\sin(x)=b$$ get $\frac{a}{1+b}+\frac{1+b}{a}$ when you add you get $\frac{a^2+(1+b)^2}{(1+b)a}$

10. misty1212

nothing cancels yet, we gotta multiply out up top $\frac{a^2+1+2b+b^2}{(1+b)a}$

11. misty1212

then since $$a^2+b^2=1$$ you have $\frac{2+2b}{(a+b)a}=\frac{2(1+b)}{(1+b)a}=\frac{2}{a}$

12. misty1212

replacing $$a$$ by $$\cos(x)$$ you are left with $\frac{2}{\cos(x)}=2\sec(x)$

13. misty1212

hope all steps are clear, it is easier to write with a and b instead of cosine and sine

14. anonymous

Thank you!! @misty1212

15. anonymous

@misty1212 when you are done can you help me??

16. misty1212

$\color\magenta\heartsuit$