1. sin(x) * (1 + tan(x))+ cos(x) * (1 + cot(x))
2. (1 - cos²(x)) * (1 + tan²(x))
#1. recall the definitions of tanx and cotx
sin(x) * (1 + sin(x)/cos(x))+ cos(x) * (1 + cos(x)/sin(x))
expand and you will find that several terms can be simplified
#2. Try these identities
1 - cos²(x) = sin²(x)
1 + tan²(x) = sec²(x)
Okay, thank you.
To simplify these expressions, we can use trigonometric identities. Let's go through each expression step by step.
1. sin(x) * (1 + tan(x)) + cos(x) * (1 + cot(x))
To simplify this expression, we'll start by applying the tangent and cotangent identities:
tan(x) = sin(x) / cos(x)
cot(x) = cos(x) / sin(x)
So, we can substitute these values into the original expression:
sin(x) * (1 + sin(x) / cos(x)) + cos(x) * (1 + cos(x) / sin(x))
Next, we'll simplify further by simplifying the fractions:
(sin(x) * cos(x) + sin(x) * sin(x)) / cos(x) + (cos(x) * sin(x) + cos(x) * cos(x)) / sin(x)
Now, we'll combine common terms in the numerator:
sin(x) * cos(x) + sin²(x) + cos(x) * sin(x) + cos²(x)
Within the second and fourth terms, sin²(x) and cos²(x), we can apply the Pythagorean identity:
sin²(x) + cos²(x) = 1
So, we can simplify further:
sin(x) * cos(x) + 1 + cos(x) * sin(x) + 1
Now, let's combine like terms:
2 + 2sin(x) * cos(x)
Therefore, the simplified expression is: 2 + 2sin(x) * cos(x).
2. (1 - cos²(x)) * (1 + tan²(x))
To simplify this expression, we'll start by applying the cosine and tangent identities:
cos²(x) = 1 - sin²(x)
tan²(x) = sin²(x) / cos²(x)
So, we can substitute these values into the original expression:
(1 - (1 - sin²(x))) * (1 + sin²(x) / (1 - sin²(x)))
Next, we'll simplify further:
(1 - 1 + sin²(x)) * (1 + sin²(x) / (1 - sin²(x)))
Now, let's combine like terms:
sin²(x) * (1 + sin²(x) / (1 - sin²(x)))
Within the fraction, we can apply the Pythagorean identity:
1 + sin²(x) = cos²(x)
So, we can simplify further:
sin²(x) * (cos²(x) / (1 - sin²(x)))
Now, let's simplify the denominator by applying the Pythagorean identity again:
1 - sin²(x) = cos²(x)
Substituting this value, we have:
sin²(x) * (cos²(x) / cos²(x))
Finally, we can cancel out the common terms:
sin²(x)
Thus, the simplified expression is: sin²(x).