How to verify this identity?sin x + tan x/ 1 + sec x = sin x   thank you very much!

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

We need to prove that :

sinx+ tanx / (1+ sec x ) = sinx

We will start from the left side.

(sinx + tanx)/ (1+ secx)

We know that tanx = sinx/cosx ans secx = 1/cosx.

Then, we will substitute with identities.

==> (sinx+ sinx/cosx) / (1+ 1/cosx)

==> [(sinx*cosx + sinx)/cosx] / [ ( cosx + 1)/cosx]

Reduce cosx.

==> (sinxcosx + sinx ) / (cosx + 1)

Now we will factor sinx from the numerator.

==> sinx*(cosx+1)/(cosx+1)

Now we will reduce cosx+1

==> sinx

Then we proved that ( sinx+tanx)/(1+secx) = sinx.

 

dstylez10's profile pic

dstylez10 | eNotes Newbie

Posted on

sin x + tan x/ 1 + sec x = sin x
Identity sec x = (tan x)/(sin x)
Factor out sin x

1. [sin x (1 + sec x)] / (1 + sec x) = sin x
Cancel 1 + sec x

2. sin x = sin x 

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, you'll have to use brackets to emphasize what is the expression that must be demonstrated.

(sin x + tan x)/ (1 + sec x) = sin x

Now, we'll multiply both sides by (1 + sec x):

(sin x + tan x) = sin x*(1 + sec x)

But sec x = 1/cos x

We'll remove the brackets from the right side:

(sin x + tan x) = sin x + sin x*(1/cos x)

But sin x/cos x = tan x

sin x + tan x = sin x + tan x

Since LHS=RHS, therefore the identity is verified.

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