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Write the left side of the identity in terms of sine and cosine. Rewrite the numerator and denominator separately. ​(Do not​ simplify.) Simplify the fraction from the previous step such that both the fractions have the common denominator The expression from the previous step then simplifies to using​ what? A. Addition and a Reciprocal Identity B. Addition and a Pythagorean Identity

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6 votes

Answer:


(a)
(2)/(cos\theta)/(1)/(sin\theta) + (4sin\theta)/(cos\theta) =6tan\theta


(b)
(2sin\theta)/(cos\theta) + (4sin\theta)/(cos\theta) =6tan\theta

(c) Addition and Quotient identity

Explanation:

Given


(2sec\theta)/(csc\theta) + (4sin\theta)/(cos\theta) =6tan\theta --- The expression missing from the question

Solving (a): Write the left hand side in terms of sin and cosine

In trigonometry:


sec\theta = (1)/(cos\theta)

and


csc\theta = (1)/(sin\theta)

So, the expression becomes:


(2 * (1)/(cos\theta))/((1)/(sin\theta)) + (4sin\theta)/(cos\theta) =6tan\theta


((2)/(cos\theta))/((1)/(sin\theta)) + (4sin\theta)/(cos\theta) =6tan\theta

Rewrite as:


(2)/(cos\theta)/(1)/(sin\theta) + (4sin\theta)/(cos\theta) =6tan\theta

Solving (b): Simplify


(2)/(cos\theta)/(1)/(sin\theta) + (4sin\theta)/(cos\theta) =6tan\theta

Change / to *


(2)/(cos\theta)*(sin\theta)/(1) + (4sin\theta)/(cos\theta) =6tan\theta


(2sin\theta)/(cos\theta) + (4sin\theta)/(cos\theta) =6tan\theta

Solving (c): The property used

To do this, we need to further simplify


(2sin\theta)/(cos\theta) + (4sin\theta)/(cos\theta) =6tan\theta

Take LCM


(2sin\theta+ 4sin\theta)/(cos\theta) =6tan\theta

Add the numerator


(6sin\theta)/(cos\theta) =6tan\theta

Apply quotient identity


(sin\theta)/(cos\theta) = tan\theta

This gives


(6sin\theta)/(cos\theta) = 6tan\theta

Hence, the properties applied are:

Addition and Quotient identity

User Kaz Yoshikawa
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