Simplify (1+tanx)^2
The answer is (1-sinx)(1+sinx)
Here's what I do:
1 + 2tanx + tan^2x
When I simplify it becomes
1 + 2(sinx/cosx) + (1+secx)(1-secx)
What am I doing wrong?
(1+tanx)^2 ≠ (1-sinx)(1+sinx)
RS = 1 - sin^2 x
= cos^2 x
and cos^2 x is certainly not equal to (1+tanx)^2
try x = 45°
(1+tan45)^2 = (1+1)^2 = 4
cos^2 (45°) = (1/√2)^2 = 1/4
no wonder you got nowhere with your work.
check the question, or check your typing.
To simplify the expression (1+tanx)^2, you can start by understanding that tanx = sinx/cosx.
So, let's substitute tanx with sinx/cosx in the expression:
(1 + sinx/cosx)^2
To simplify further, we can expand the square of a binomial. Applying the formula (a+b)^2 = a^2 + 2ab + b^2, we get:
1^2 + 2(1)(sinx/cosx) + (sinx/cosx)^2
Simplifying the expression, we have:
1 + 2sinx/cosx + sin^2x/cos^2x
Now, we need to convert sinx and cosx into their equivalent forms using trigonometric identities.
Using the Pythagorean identity sin^2x + cos^2x = 1, we can write cos^2x as 1 - sin^2x:
1 + 2sinx/cosx + sin^2x/(1 - sin^2x)
Next, we can factor out sinx from the numerator of the third term:
1 + 2sinx/cosx + sinx*sinx/(1 - sin^2x)
Simplifying further by canceling out the common factor sinx, we have:
1 + 2sinx/cosx + sinx/(1 - sinx)(1 + sinx)
Now, we can rewrite cosx as 1/sinx using the reciprocal identity:
1 + 2sinx/(1/sinx) + sinx/(1 - sinx)(1 + sinx)
Simplifying the expression inside the brackets:
1 + 2sin^2x + sinx/(1 - sinx)(1 + sinx)
Finally, we can simplify the expression by factoring out a common factor (1 - sinx) from the denominator:
(1 - sinx) + 2sin^2x + sinx/(1 - sinx)(1 + sinx)
This can be further simplified:
(1 - sinx)(1 + 2sinx) + sinx/(1 - sinx)(1 + sinx)
So, the simplified form of (1+tanx)^2 is (1 - sinx)(1 + 2sinx) + sinx/(1 - sinx)(1 + sinx).
The answer you provided, (1 - sinx)(1 + sinx), is incorrect because you missed the term 2sinx.