use basic identities to simplify the expression.
csc θ cot θ / sec θ
To simplify the expression csc θ cot θ / sec θ, we can start by using the reciprocal identities:
1. csc θ = 1 / sin θ
2. cot θ = 1 / tan θ
3. sec θ = 1 / cos θ
Substituting these values into the expression, we get:
(1 / sin θ) * (1 / (1 / tan θ)) / (1 / cos θ)
Next, we can simplify by multiplying the numerators and denominators:
(1 / sin θ) * (tan θ / 1) * (cos θ / 1)
This simplifies to:
tan θ * cos θ / sin θ
Finally, since tan θ = sin θ / cos θ, we can substitute this value into the expression:
(sin θ / cos θ) * cos θ / sin θ
The sin θ cancels out with one of the sin θ terms, and the cos θ cancels out with one of the cos θ terms, resulting in:
1
So the simplified expression is 1.
To simplify the expression csc θ cot θ / sec θ using basic trigonometric identities, we can break down each trigonometric function into its constituent parts and simplify from there.
1. Start by substituting the reciprocals of csc θ and sec θ, which are sin θ and cos θ respectively:
csc θ = 1 / sin θ
sec θ = 1 / cos θ
2. Substitute the reciprocal of cot θ, which is 1 / tan θ:
cot θ = 1 / tan θ
Now the expression becomes:
(1 / sin θ)(1 / tan θ) / (1 / cos θ)
3. Simplify by multiplying the numerator and denominator by the reciprocal of the denominator:
(1 / sin θ)(1 / tan θ) x (cos θ / 1)
Simplify further by canceling out common factors:
(1 / sin θ) x (cos θ / tan θ)
4. Rewrite the remaining trigonometric functions in terms of sine and cosine using the trigonometric identity tan θ = sin θ / cos θ:
(1 / sin θ) x (cos θ / (sin θ / cos θ))
Simplify by canceling out the common factors:
(1 / sin θ) x (cos θ x cos θ / sin θ)
5. Multiply the numerators and denominators:
(cos θ x cos θ) / (sin θ x sin θ)
6. Finally, use the Pythagorean Identity sin^2 θ + cos^2 θ = 1 to simplify further:
(cos^2 θ) / (sin^2 θ)
Therefore, the simplified expression is: cos^2 θ / sin^2 θ.