establish the identity

[justseel]

My math teacher ask me to establish the identity in two ways
from right side to the left side
and from left ti right
sec=sin/cos + cos/1+sin

 

[Deleted member 4993]

My math teacher ask me to establish the identity in two ways
from right side to the left side
and from left ti right
sec=sin/cos + cos/1+sin

What have you tried?

What are your thoughts?

Please share your work with us ...

If you are stuck at the beginning tell us and we'll start with the definitions.

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[justseel]

yes I,m stuck at the beginning
i tired to do this but i didnt know how end it
1/cosx = sinx+1/cosxsinxcosx

 

[richardt]

Greetings:

A hint: Try multiplying numerator and denominator of cosx/(1+sinx) by the conjugate of 1 + sinx.

Rich

 

[Deleted member 4993]

Enough time has gone by: Right to left

\(\displaystyle \dfrac{sin(\theta)}{cos(\theta)} + \dfrac{cos(\theta)}{1+sin(\theta)}\)

\(\displaystyle = \ \dfrac{sin(\theta)*[1+sin(\theta)] + cos^2(\theta)}{cos(\theta)*[1+sin(\theta)]} \)

\(\displaystyle = \ \dfrac{1+sin(\theta)}{cos(\theta)*[1+sin(\theta)]} \)

\(\displaystyle = \ \dfrac{1}{cos(\theta)} \)

\(\displaystyle = \ sec(\theta) \)

Now just go back-wards.... for left-to-right