Simplify each expression using fundamental identities.
1. Sin theta cot theta
2. 1-sin^2 theta / cos theta
Use:
cot(θ)=cos(θ)/sin(θ)
sin²(θ)+cos²(θ)=1
To simplify each expression using fundamental identities, let's go step by step.
1. Sin theta cot theta:
Recall the fundamental identity for cotangent (cot): cot theta equals 1 divided by tan theta.
So we can rewrite the expression as sin theta multiplied by 1 divided by tan theta.
Now, the fundamental identity for tangent (tan) is tan theta equals sin theta divided by cos theta.
Therefore, the expression becomes sin theta multiplied by 1 divided by (sin theta divided by cos theta).
To divide by a fraction, we can multiply by its reciprocal. So, the expression simplifies to sin theta multiplied by cos theta divided by sin theta.
Finally, the sin theta on the numerator and denominator cancels out, leaving us with just cos theta as the simplified expression.
2. 1-sin^2 theta / cos theta:
For this expression, we start by simplifying the numerator. We have (1 - sin^2 theta). This should look familiar as the Pythagorean Identity: sin^2 theta + cos^2 theta equals 1.
Using the Pythagorean Identity, we can substitute 1 minus cos^2 theta for sin^2 theta. So, the numerator becomes 1 - cos^2 theta.
The expression now becomes (1 - cos^2 theta) divided by cos theta.
Next, we can factor the numerator using the difference of squares formula. The difference of squares formula states that a^2 - b^2 equals (a + b)(a - b).
Applying this formula to our numerator, we get (1 + cos theta)(1 - cos theta) divided by cos theta.
Finally, we can cancel out the common factor of (1 - cos theta) in both the numerator and denominator, leaving us with just (1 + cos theta) as the simplified expression.