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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]
==> (sinxcosx + sinx ) / (cosx + 1)
Now we will factor sinx from the numerator.
Now we will reduce cosx+1
Then we proved that ( sinx+tanx)/(1+secx) = sinx.
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
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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