Simplify each trigonometric expression
a) Sin(theta)Cot(theta)-Sin(theta)Cos(theta)
c)(sin x + cos x)(sin x - cos x)+ 2 cos(squared)x
sinØcotØ - sinØcosØ
= sinØcosØ/sinØ - sinØcosØ
= cosØ - sinØcosØ
= cosØ(1-sinØ)
(sin x + cos x)(sin x - cos x)+ 2 cos(squared)x
= sin^2 x - cos^2 x + 2cos^2 x
= sin^2x + cos^2x
= 1
a) To simplify the expression Sin(theta)Cot(theta) - Sin(theta)Cos(theta), we can use the trigonometric identities:
Cot(theta) = Cos(theta) / Sin(theta)
Therefore, we can rewrite the expression as:
Sin(theta)(Cos(theta) / Sin(theta)) - Sin(theta)Cos(theta)
Simplifying further, we can cancel out the Sin(theta) terms:
Cos(theta) - Sin(theta)Cos(theta)
Now, we can factor out the common factor of Cos(theta):
Cos(theta)(1 - Sin(theta))
So, the simplified expression is Cos(theta)(1 - Sin(theta)).
c) To simplify the expression (sin x + cos x)(sin x - cos x) + 2cos^2(x), we can use the trigonometric identity:
sin^2(x) - cos^2(x) = 1
First, let's expand the expression:
(sin x + cos x)(sin x - cos x) + 2cos^2(x)
= sin^2(x) - cos^2(x) + 2cos^2(x)
Using the trigonometric identity, we substitute sin^2(x) - cos^2(x) with 1:
1 + 2cos^2(x)
Now, we can simplify further by combining like terms:
1 + 2cos^2(x) = 2cos^2(x) + 1
So, the simplified expression is 2cos^2(x) + 1.