verify the identity:
(cosx/1+sinx) + (1+sinx/cosx) = 2secx
sinxcosx/cos^2x-sin^2x=tanx/1-tan^2x
1st:
putting all over a common denominator,
= (cos^2 + (1+sin)^2)/(cos(1+sin))
= (cos^2 + 1 + 2sin + sin^2)/(cos(1+sin))
= (2 + 2sin)/(cos(1+sin))
= 2/cos
= 2sec
2nd:
sin2x/cos2x = tan2x
done
To verify the identity, we need to simplify both sides of the equation separately and check if they are equal to each other.
Starting with the left side of the equation:
(cosx/1+sinx) + (1+sinx/cosx)
To simplify this expression, we need to find a common denominator. The common denominator in this case will be (1 + sinx)(cosx).
So, the expression becomes:
[(cosx)(cosx) + (1 + sinx)(1 + sinx)] / [(1 + sinx)(cosx)]
Expanding and simplifying the numerator, we have:
[cos^2(x) + 1 + 2sinx + sin^2(x)] / [(1 + sinx)(cosx)]
Next, we can combine the terms in the numerator:
[2 + 2sinx + sin^2(x) + cos^2(x)] / [(1 + sinx)(cosx)]
Knowing that cos^2(x) + sin^2(x) = 1 (from the Pythagorean identity), the expression further simplifies to:
[2 + 2sinx + 1] / [(1 + sinx)(cosx)]
Simplifying the numerator gives:
[3 + 2sinx] / [(1 + sinx)(cosx)]
Now, let's simplify the right side of the equation, which is 2secx.
Recall that secx = 1/cosx, so 2secx = 2/cosx.
Combining the expressions, we have:
[3 + 2sinx] / [(1 + sinx)(cosx)] = 2/cosx
We can simplify further by multiplying both sides of the equation by [(1 + sinx)(cosx)]:
[3 + 2sinx] = 2(1 + sinx)
Expanding and simplifying:
3 + 2sinx = 2 + 2sinx
We can see that both sides of the equation are indeed equal. Hence, the left side of the equation (cosx/1+sinx) + (1+sinx/cosx) equals the right side of the equation 2secx.
Therefore, we have verified the identity.