how do i solve (1/sin[theta]+1)+(1/csc[theta]+1)=1 ?

[cinimini28]

i dont understand how to solve this problem please help.

(1/sin[theta]+1)+(1/csc[theta]+1)=1

 

[soroban]

Hello, cinimini28!

I don't think you mean "solve" . . . It's an identity.

\(\displaystyle \frac{1}{\sin\theta+1}+\frac{1}{\csc\theta+1} \:=\:1\)

\(\displaystyle \text{On the left, the second fraction is: }\;\frac{1}{\frac{1}{\sin\theta} + 1}\)

\(\displaystyle \text{Multiply by }\frac{\sin\theta}{\sin\theta}\!:\quad \frac{\sin\theta}{\sin\theta}\cdot \frac{1}{\frac{1}{\sin\theta} + 1} \;=\;\frac{\sin\theta}{1 + \sin\theta}\)


\(\displaystyle \text{The left side becomes: }\:\frac{1}{1 + \sin\theta } + \frac{\sin\theta}{1 + \sin\theta} \;=\;\frac{1+\sin\theta}{1+\sin\theta} \;=\;1\)