Write the trigonometric expression in terms of sine and cosine, and then simplify.
1). (csc θ − sin θ)/(cos θ)
____________.
2). Simplify the trigonometric expression.
(cos u + 1)/(sin u) +
(sin u)/(1 + cos u)
______________.
(csc θ − sin θ)/(cos θ)
= (1/sinθ - sinθ)/cosθ
= (1-sin^2θ)/(sinθ cosθ)
= cos^2θ/(sinθ cosθ)
= cosθ/sinθ
= cotθ
1) To write the trigonometric expression in terms of sine and cosine, we can make use of the reciprocal identities:
csc θ = 1/sin θ
cos θ = 1/csc θ
So, the expression becomes:
(1/sin θ - sin θ)/(cos θ)
Now, recall that sin θ = 1/csc θ. Substituting this in the expression, we have:
(1/(1/csc θ) - sin θ)/(cos θ)
Simplifying further:
(csc θ - sin θ)/(cos θ)
2) To simplify the expression:
(cos u + 1)/(sin u) + (sin u)/(1 + cos u)
Let's combine the fractions with a common denominator of sin u(1 + cos u):
[(cos u + 1)(1 + cos u) + (sin u)(sin u)] / [sin u(1 + cos u)]
Expanding the numerator:
[cos u + cos^2 u + 1 + cos u + sin^2 u] / [sin u(1 + cos u)]
Simplifying:
[2cos u + cos^2 u + sin^2 u + 1] / [sin u(1 + cos u)]
Now, recall the trigonometric identity:
1 + cos^2 u = sin^2 u
Substituting this in the expression:
[2cos u + sin^2 u + sin^2 u + 1] / [sin u(1 + cos u)]
Simplifying further:
[2cos u + 2sin^2 u + 1] / [sin u(1 + cos u)]
1). To express the trigonometric expression (csc θ − sin θ)/(cos θ) in terms of sine and cosine, we can rewrite csc θ as 1/sin θ.
So, (csc θ − sin θ)/(cos θ) becomes (1/sin θ - sin θ)/(cos θ).
Next, we want to combine the fractions. To do this, we need a common denominator. The common denominator in this case is sin θ.
Therefore, (1/sin θ - sin θ)/(cos θ) becomes ((1 - sin^2 θ)/sin θ)/(cos θ).
Using the Pythagorean Identity sin^2 θ + cos^2 θ = 1, we can rewrite the numerator as (cos^2 θ)/sin θ.
So, ((cos^2 θ)/sin θ)/(cos θ) simplifies to (cos^2 θ)/(cos θ * sin θ).
Now, we can cancel out the cos θ term in the denominator, leaving us with cos θ / sin θ.
Since cos θ / sin θ is equal to cot θ, the simplified expression is cot θ.
2). To simplify the trigonometric expression (cos u + 1)/(sin u) + (sin u)/(1 + cos u), we need to find a common denominator.
The common denominator in this case is (1 + cos u).
So, we rewrite the expression as ((cos u + 1)(1 + cos u) + (sin u)(sin u))/(sin u * (1 + cos u)).
Expanding the numerator, we have (cos u + 1 + cos^2 u + cos u + sin^2 u)/(sin u * (1 + cos u)).
Simplifying the numerator, we have (2cos u + 1 + cos^2 u + sin^2 u)/(sin u * (1 + cos u)).
Using the Pythagorean Identity sin^2 u + cos^2 u = 1, we can simplify further to have (2cos u + 1 + 1)/(sin u * (1 + cos u)).
This simplifies to (2cos u + 2)/(sin u * (1 + cos u)).
Factoring out a 2 from the numerator, we have 2(cos u + 1)/(sin u * (1 + cos u)).
Finally, we can cancel out the (cos u + 1) term in the numerator and denominator, leaving us with just 2/sin u.
Since 2/sin u is equal to 2csc u, the simplified expression is 2csc u.
(csc θ − sin θ)/(cos θ)
(1/sin- sin^2/sin)/cos
(1-sin^2)*sin/cos
cos^2*sin/cos
cos*sin
:(cos+1)/sin + sin/(1+cosu)
common denominaator:sin(1+cos)
1+2cos+cos^2 + sin^2 all over denominator
2(1+cos)/sin(1+cos)
2/sinU
check that.