secx+tanx/secx- tanx = 1+2sinx+sin^2x/ cos^2x
I will assume you want to prove the identity to be true.
The way you typed , the statement is NOT an identity
You must have meant:
(secx+tanx)/(secx- tanx) = (1+2sinx+sin^2x)/ cos^2x
LS = (1/cosx + sinx/cosx) / (1/cosx - sinx/cosx)
= ( (1+sinx)/cosx)/( (1-sinx)/cosx)
= (1+sinx) / (1 - sinx)
RS = (1+sinx)^2 / (1 - sin^2 x)
= (1+sinx)(1+sinx)/( (1-sinx)(1+sinx))
= (1+sinx)/(1-sinx)
= LS
To simplify the expression secx+tanx/secx-tanx, we can start by rationalizing the denominator.
We're given the expression: secx+tanx/secx-tanx
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is secx + tanx:
(secx+tanx/secx-tanx) * (secx+tanx/secx+tanx)
When we multiply the numerator and denominator, we get:
(sec^2x + secx*tanx + secx*tanx + tan^2x) / (sec^2x - tan^2x)
This simplifies to:
(sec^2x + 2secx*tanx + tan^2x) / (sec^2x - tan^2x)
Now, we can apply the trigonometric identity:
sec^2x - tan^2x = 1
Substituting this identity in, we have:
(sec^2x + 2secx*tanx + tan^2x) / 1
Simplifying further, we get:
sec^2x + 2secx*tanx + tan^2x
Next, let's simplify the numerator of the expression 1 + 2sinx + sin^2x:
Using the identity cos^2x + sin^2x = 1, we can rewrite sin^2x as 1 - cos^2x:
1 + 2sinx + sin^2x = 1 + 2sinx + (1 - cos^2x)
Combining like terms, we have:
2 + 2sinx - cos^2x
Now, let's simplify cos^2x. Using the identity sin^2x + cos^2x = 1, we can rewrite cos^2x as 1 - sin^2x:
2 + 2sinx - cos^2x = 2 + 2sinx - (1 - sin^2x)
Distributing the negative sign, we get:
2 + 2sinx - 1 + sin^2x
Combining like terms, we have:
1 + 2sinx + sin^2x
Comparing this to the numerator of our rationalized expression, we can see that they are equal.
Therefore, we can conclude that:
(secx+tanx/secx-tanx) = (1+2sinx+sin^2x) / cos^2x
To simplify the expression secx+tanx/secx-tanx, we can use the identities:
- secx = 1/cosx
- tanx = sinx/cosx
Substituting these values into the expression, we get:
(1/cosx + sinx/cosx) / (1/cosx - sinx/cosx)
Now, to simplify further, we can combine the fractions by multiplying the numerator and denominator of the big fraction by cosx. This yields:
((1 + sinx) * cosx) / ((1 - sinx) * cosx)
Now, we can divide cosx out in the numerator and denominator:
(1 + sinx) / (1 - sinx)
Next, let's simplify the right side of the equation. We can rewrite the expression 1 + 2sinx + sin^2x / cos^2x as:
(cos^2x + 2sinx*cosx + sin^2x) / cos^2x
Now, simplifying the numerator a bit, we have:
(1 + sinx)^2 / cos^2x
Using the identity sin^2x + cos^2x = 1, we can rewrite cos^2x as 1 - sin^2x:
(1 + sinx)^2 / (1 - sin^2x)
Applying the identity (a + b)^2 = a^2 + 2ab + b^2, we can expand (1 + sinx)^2:
(1 + 2sinx + sin^2x) / (1 - sin^2x)
Now, we can observe that the numerator and denominator of the right side are the same as in the previous expression.
Therefore, we've shown that the left side of the equation (secx+tanx/secx-tanx) is equal to the right side of the equation (1 + 2sinx + sin^2x / cos^2x).