cos(+pi/2-theta)/csctheta+cos^2theta
cos(π/2 - Ø) = sinØ , complementary angle property
then
cos(+pi/2-theta)/csctheta+cos^2theta
= cos(π/2-Ø) * 1/cscØ + cos^2 Ø
= sinØ ( sinØ) + cos^2 Ø
= sin^2 Ø + cos^2 Ø
= 1
To simplify the expression cos(+pi/2-theta)/csctheta + cos^2theta, we can start by simplifying the individual trigonometric functions and then combining the terms.
Let's break it down step by step:
1. Simplifying cos(+pi/2-theta):
Using the identity cos(A - B) = cos(A)cos(B) + sin(A)sin(B), we can rewrite the expression as:
cos(+pi/2-theta) = cos(pi/2)cos(-theta) + sin(pi/2)sin(-theta)
= (0)cos(-theta) + (1)sin(-theta)
= sin(-theta)
Since sin function is an odd function (sin(-x) = -sin(x)), we have:
sin(-theta) = -sin(theta)
Therefore, cos(+pi/2-theta) simplifies to -sin(theta).
2. Simplifying csctheta:
The reciprocal of sin(theta) is called csc(theta), so:
csctheta = 1/sin(theta)
3. Now, let's substitute the simplified expressions back into the original expression:
(-sin(theta))/(1/sin(theta)) + cos^2(theta)
Multiplying by the reciprocal of 1/sin(theta) is equivalent to multiplying by sin(theta):
-sin(theta) * sin(theta) + cos^2(theta)
Simplifying further:
-sin^2(theta) + cos^2(theta)
Finally, we can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 to simplify further:
-sin^2(theta) + cos^2(theta) = -1 + 1
= 0
Therefore, the simplified form of the expression cos(+pi/2-theta)/csctheta + cos^2(theta) is 0.