Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
I assumed that
(1 + tan^2 u)(1 - sin^2 u) = 1
Note that (1 + tan^2 u) = sec^2 u. [pythagorean identity]
And that (1 - sin^2 u) = cos^2 u. [pythagorean identity, from cos^2 u + sin^2 u = 1]
Thus,
( sec^2 u )( cos^2 u )= 1
Note that sec u = 1 / cos u,
(1 / cos u)^2 (cos^2 u) = 1
cos^2 u / cos^2 u = 1
1 = 1
Yeah I had this answer , just making sure , Thank you :)
To verify the given identity, let's start by simplifying both sides of the equation separately:
Left-hand side (LHS):
(1 + tan^2(u))(1 - sin^2(u))
Recall the trigonometric identity: tan^2(u) = sec^2(u) - 1.
Using this identity, we can rewrite the first part of the LHS.
(1 + (sec^2(u) - 1))(1 - sin^2(u))
Simplifying, we get:
(sec^2(u))(1 - sin^2(u))
Recall another trigonometric identity: sec^2(u) = 1 + tan^2(u).
Using this identity, we can rewrite the first part of the expression.
(1 + tan^2(u))(1 - sin^2(u))
Now let's simplify the second part:
(1 - sin^2(u))
Using the identity sin^2(u) + cos^2(u) = 1, we can rewrite the expression:
cos^2(u)
Therefore, the LHS becomes:
(1 + tan^2(u))(cos^2(u))
Now, let's simplify the right-hand side (RHS):
1
Now we can compare the simplified LHS and the RHS:
(1 + tan^2(u))(cos^2(u)) = 1
Thus, we have verified that the identity (1 + tan^2(u))(1 - sin^2(u)) = 1 holds true.
To verify the given identity: (1 + tan^2u)(1 - sin^2u) = 1, we can simplify the left side of the equation and show that it equals 1.
Let's start by expanding the left side using the identity: 1 - sin^2(u) = cos^2(u).
(1 + tan^2(u))(1 - sin^2(u)) = 1
(1 + tan^2(u))(cos^2(u)) = 1
Now, expand the expression on the left side using the identity: tan^2(u) = sec^2(u) - 1.
(cos^2(u) * (sec^2(u) - 1)) = 1
Applying the distributive property, we get:
(cos^2(u) * sec^2(u)) - cos^2(u) = 1
Next, use the identity: sec^2(u) = 1 + tan^2(u).
(cos^2(u) * (1 + tan^2(u))) - cos^2(u) = 1
Now, distribute the cos^2(u):
(cos^2(u) + cos^2(u) * tan^2(u)) - cos^2(u) = 1
Canceling the "cos^2(u)" terms:
cos^2(u) * tan^2(u) = 1
To simplify further, we can use the identity: tan^2(u) = sec^2(u) - 1.
cos^2(u) * (sec^2(u) - 1) = 1
Now, we expand the expression using the distributive property:
cos^2(u) * sec^2(u) - cos^2(u) = 1
Finally, applying the identity: sec^2(u) = 1 + tan^2(u), we substitute it back in:
cos^2(u) * (1 + tan^2(u)) - cos^2(u) = 1
Again, canceling the "cos^2(u)" terms:
cos^2(u) + cos^2(u) * tan^2(u) - cos^2(u) = 1
The "cos^2(u)" terms cancel out:
cos^2(u) * tan^2(u) = 1
This proves that the left side of the equation simplifies to 1. Hence, the identity has been verified.