Markadam1
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Use basic identities to simplify the expression.

Cos^2 0/sin^2 0 +csc 0 sin 0

A. Csc^2 0
B. Sec^2 0
C. 1
D. Tan^2 0
Look at the picture

Use basic identities to simplify the expression Cos2 0sin2 0 csc 0 sin 0 A Csc2 0 B Sec2 0 C 1 D Tan2 0 Look at the picture class=

Respuesta :

semsee45

Answer:

[tex]\textsf{A)}\quad \csc^2\theta[/tex]

Step-by-step explanation:

Given trigonometric expression:

[tex]\dfrac{\cos^2\theta}{\sin^2\theta}+\csc\theta\sin\theta[/tex]

As csc(θ) is the reciprocal of sin(θ), replace csc(θ) with 1/sin(θ):

[tex]\dfrac{\cos^2\theta}{\sin^2\theta}+\dfrac{1}{\sin\theta}\cdot \sin\theta[/tex]

Therefore:

[tex]\dfrac{\cos^2\theta}{\sin^2\theta}+\dfrac{\sin\theta}{\sin\theta}[/tex]

Multiply the second fraction by sin(θ)/sin(θ) so that both fractions have the same denominator:

[tex]\dfrac{\cos^2\theta}{\sin^2\theta}+\dfrac{\sin\theta\cdot \sin\theta}{\sin\theta\cdot \sin\theta}[/tex]

[tex]\dfrac{\cos^2\theta}{\sin^2\theta}+\dfrac{\sin^2\theta}{\sin^2\theta}[/tex]

Combine the fractions:

[tex]\dfrac{\cos^2\theta+\sin^2\theta}{\sin^2\theta}[/tex]

Using the Pythagorean identity sin⁡⁡²θ + cos²θ = 1, we can simplify further:

[tex]\dfrac{1}{\sin^2\theta}[/tex]

As csc(θ) is the reciprocal of sin(θ), then:

[tex]\csc^2\theta[/tex]

msm555

Answer:

A. [tex] \csc^2 \theta [/tex]

Step-by-step explanation:

To simplify the expression [tex] \dfrac{\cos^2 \theta}{\sin^2 \theta} + \csc \theta \sin \theta [/tex], we will use the basic trigonometric identities.

First, let's rewrite [tex] \csc \theta [/tex] in terms of sine:

[tex] \csc \theta = \dfrac{1}{\sin \theta} [/tex]

Now, we rewrite the expression:

[tex] \dfrac{\cos^2 \theta}{\sin^2 \theta} + \csc \theta \sin \theta = \dfrac{\cos^2 \theta}{\sin^2 \theta} + \dfrac{1}{\sin \theta} \cdot \sin \theta [/tex]

Using the fact that [tex] \sin \theta \cdot \csc \theta = 1 [/tex], we have:

[tex] \dfrac{\cos^2 \theta}{\sin^2 \theta} + 1 [/tex]

Now, we can express [tex] \dfrac{\cos^2 \theta}{\sin^2 \theta} [/tex] using the Pythagorean identity [tex] \cos^2 \theta = 1 - \sin^2 \theta [/tex]:

[tex] \dfrac{1 - \sin^2 \theta}{\sin^2 \theta} + 1 [/tex]

Now, we can simplify:

[tex] \dfrac{1}{\sin^2 \theta} - \dfrac{\sin^2 \theta}{\sin^2 \theta} + 1 [/tex]

[tex] \csc^2 \theta - 1 + 1 [/tex]

[tex] \csc^2 \theta [/tex]

So, the simplified expression is [tex] \csc^2 \theta [/tex]. Therefore, the answer is:

A. [tex] \csc^2 \theta [/tex]

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