The first step in solving this equation is to distribute :

At this point, simplify using known Pythagorean identities. The left quantity simplifies such:

and the right quantity simplifies such:

Thus, we end up with:

,

which our Pythagorean identity tells us is equivalent to .

Thus,

First we expand the inverse identities into fractional form:

Invert the bottom fraction and distribute into the top, keeping track of the negative:

Using the Pythagorean identity , our equation becomes:

At this point, cross-cancel to obtain .

In order to solve , first identify all the recognizable identities.

The identities that can be used for this scenario are:

The first identity can also be manipulated.

Replace  with the correct identities and simplify.

We can multiply the bottom with its conjugate and obtain:

Then we can use the pythagorean identity for the cosines and sines:

Finally, we can split the fractions up and translate them into the trigonometric identity:

Alternatively, you could take this and other answer choices and work the opposite way by translating all of the trigonometric ratios into sines and cosines, using the identities.

Another potential option is to try certain angle values for the original identity and selecting the choice that most closely matches the answer choices when the same angle is substituted (although one must beware of where the function(s) are undefined).

To answer this question, use the sum formula for sine:

In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:

Now we notice that we can further group the terms:

The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:

This reduced simply to the sum function for sine:

In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of  , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:

Where  and  are integers. If we work to get a common denominator of 12:

We can now reduce the equation to obtain a simpler form:

Using this equation it is simple to see that  and  and the operation is subtraction. Therefore rewriting the equation:

Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:

We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.

This reduces to:

In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of  , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:

Where  and  are integers. If we work to get a common denominator of 12:

We can now reduce the equation to obtain a simpler form:

Using this equation it is simple to see that  and  and the operation is addition. Therefore rewriting the equation:

Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:

We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.

This reduces to:

In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of  , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:

Where  and  are integers. If we work to get a common denominator of 12:

We can now reduce the equation to obtain a simpler form:

Using this equation we actually need to flip the fractions as follows:

Now we can get the right values of;  and  and the operation is subtraction. Therefore rewriting the equation:

Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:

We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.

This reduces to:

Finally, we obtain the solution:

The key here is to use this sum/product identity:

In this case, and . Note as well that because subtracts, it will translate into . So using the identity, we can state as...

Bear in mind that for a sine function of form

...our phase shift is equal to...