rebekahlm

13.01.2021 •

Mathematics

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suselygonza

22.02.2022

Mathematics

(a) $\frac{2}{cos\theta}/\frac{1}{sin\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$

(b) $\frac{2sin\theta}{cos\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$

Step-by-step explanation:

Given

$\frac{2sec\theta}{csc\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$ --- The expression missing from the question

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

In trigonometry:

$sec\theta = \frac{1}{cos\theta}$

and

$csc\theta = \frac{1}{sin\theta}$

So, the expression becomes:

$\frac{2 * \frac{1}{cos\theta}}{\frac{1}{sin\theta}} + \frac{4sin\theta}{cos\theta} =6tan\theta$

$\frac{\frac{2}{cos\theta}}{\frac{1}{sin\theta}} + \frac{4sin\theta}{cos\theta} =6tan\theta$

Rewrite as:

$\frac{2}{cos\theta}/\frac{1}{sin\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$

Solving (b): Simplify

$\frac{2}{cos\theta}/\frac{1}{sin\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$

Change / to *

$\frac{2}{cos\theta}*\frac{sin\theta}{1} + \frac{4sin\theta}{cos\theta} =6tan\theta$

$\frac{2sin\theta}{cos\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$

Solving (c): The property used

To do this, we need to further simplify

$\frac{2sin\theta}{cos\theta} + \frac{4sin\theta}{cos\theta} =6tan\theta$

Take LCM

$\frac{2sin\theta+ 4sin\theta}{cos\theta} =6tan\theta$

Add the numerator

$\frac{6sin\theta}{cos\theta} =6tan\theta$

Apply quotient identity

$\frac{sin\theta}{cos\theta} = tan\theta$

This gives

$\frac{6sin\theta}{cos\theta} = 6tan\theta$

Hence, the properties applied are:

Addition and Quotient identity

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