Simplify each expression using fundamental identities.
1. Sin theta cot theta
2. 1-sin^2 theta / cos theta
1.
sinØcotØ
= sinØ(cosØ/sinØ)
= cosØ
2.
( 1 - sin^2Ø)/cosØ
= cos^2 Ø/cosØ
= cosØ
To simplify the given expressions using fundamental identities, we will use the basic trigonometric identities.
1. To simplify sin(theta) * cot(theta), we will rewrite cot(theta) as cos(theta)/sin(theta).
Therefore, sin(theta) * cot(theta) = sin(theta) * cos(theta) / sin(theta).
Since sin(theta) appears in both the numerator and denominator, we can cancel them out:
sin(theta) * cos(theta) / sin(theta) = cos(theta).
Therefore, sin(theta) * cot(theta) simplifies to cos(theta).
2. To simplify 1 - sin^2(theta) / cos(theta), we will use the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.
By substituting 1 - sin^2(theta) with cos^2(theta) from the Pythagorean identity, we get:
1 - sin^2(theta) = cos^2(theta).
Substituting this into the expression, we have:
1 - sin^2(theta) / cos(theta) = cos^2(theta) / cos(theta).
Since cos(theta) appears in both the numerator and denominator, we can cancel them out:
cos^2(theta) / cos(theta) = cos(theta).
Therefore, 1 - sin^2(theta) / cos(theta) simplifies to cos(theta).