Final answer:
To simplify (1+tan²x)sin²x, we use the Pythagorean identity to realize that 1+tan²x equals sec²x. After replacing the expression with sec²xsin²x, we find that it simplifies to tan²x, which is the final answer.
Step-by-step explanation:
The question involves simplifying a trigonometric expression using trigonometric identities: (1+tan²x)sin²x. To simplify this expression, we can make use of the Pythagorean identity: tan²x + 1 = sec²x, which tells us that 1 + tan²x is equal to sec²x.
Firstly, let's replace (1 + tan²x) with sec²x:
- (1 + tan²x)sin²x = sec²xsin²x
Now, knowing that secx = 1/cosx and using the identity sin²x + cos²x = 1, we can rewrite sec²x as 1/cos²x:
- sec²xsin²x = (1/cos²x)sin²x = sin²x/cos²x
Finally, we recognize that sin²x/cos²x is the definition of tan²x:
Therefore, the simplified form of the original expression is tan²x.