Question

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

Answer 1

`(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