Answer:
[tex](a)[/tex] [tex]\frac{2}{cos\theta}/\frac{1}{sin\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
[tex](b)[/tex] [tex]\frac{2sin\theta}{cos\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
(c) Addition and Quotient identity
Step-by-step explanation:
Given
[tex]\frac{2sec\theta}{csc\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex] --- The expression missing from the question
Solving (a): Write the left hand side in terms of sin and cosine
In trigonometry:
[tex]sec\theta = \frac{1}{cos\theta}[/tex]
and
[tex]csc\theta = \frac{1}{sin\theta}[/tex]
So, the expression becomes:
[tex]\frac{2 * \frac{1}{cos\theta}}{\frac{1}{sin\theta}} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
[tex]\frac{\frac{2}{cos\theta}}{\frac{1}{sin\theta}} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
Rewrite as:
[tex]\frac{2}{cos\theta}/\frac{1}{sin\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
Solving (b): Simplify
[tex]\frac{2}{cos\theta}/\frac{1}{sin\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
Change / to *
[tex]\frac{2}{cos\theta}*\frac{sin\theta}{1} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
[tex]\frac{2sin\theta}{cos\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
Solving (c): The property used
To do this, we need to further simplify
[tex]\frac{2sin\theta}{cos\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta[/tex]
Take LCM
[tex]\frac{2sin\theta+ 4sin\theta}{cos\theta} =6tan\theta[/tex]
Add the numerator
[tex]\frac{6sin\theta}{cos\theta} =6tan\theta[/tex]
Apply quotient identity
[tex]\frac{sin\theta}{cos\theta} = tan\theta[/tex]
This gives
[tex]\frac{6sin\theta}{cos\theta} = 6tan\theta[/tex]
Hence, the properties applied are:
Addition and Quotient identity
