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.

Question

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.

misty1212 · Answer

HI!!

misty1212 · Answer

lets see if we can write this in math ok?

misty1212 · Answer

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

anonymous · Answer

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

misty1212 · Answer

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

anonymous · Answer

Isn't it 2sec x

misty1212 · Answer

yes it is

anonymous · Answer

because everything else cancels out. yay

misty1212 · Answer

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 
$$\frac{a^2+(1+b)^2}{(1+b)a}$$

misty1212 · Answer

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

misty1212 · Answer

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

misty1212 · Answer

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

misty1212 · Answer

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

anonymous · Answer

Thank you!! @misty1212

anonymous · Answer

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

misty1212 · Answer

$$\color\magenta\heartsuit$$