SOLUTION: Show that the equation sec x + sqrt3 cosec x = 4 can be written in the form sin x + sqrt3 cos x = 2 sin 2x

Question 826527: Show that the equation sec x + sqrt3 cosec x = 4 can be written in the form
sin x + sqrt3 cos x = 2 sin 2x
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Looking at what we have and where we want to end up, , we should notice some things:

There are no sec's or csc's at the end. So somehow we need to get rid of them.
Since sin(2x) = 2sin(x)cos(x), turning everything into sin's and cos's looks promising.

Putting these together, along with the facts that a) sec and csc are reciprocals of cos and sin, respectively; and b) the product of reciprocals is always a 1, multiplying both sides by sin(x)cos(x) will so a lot of good:

On the left side we need to use the Distributive Property:

The reciprocal Trig functions will cancel each other out (since their product is 1):

If we use the Commutative Property on the second term, out left side is exactly what we wanted it to be:

Now we need to fix up the right side (without changing the left side). As we pointed out earlier, sin(2x) = 2sin(x)cos(x). If we factor a 2 out of the left side we get:

And we can now substitute in the sin(2x):

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