The first step to simplifying is to remember an important trig identity.

If we rewrite it to look like the denominator, it is.

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

The first step to simplifying is to remember an important trig identity.

If we rewrite it to look like the denominator, it is.

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

The first step to simplifying is to remember an important trig identity.

If we rewrite it to look like the denominator, it is.

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

The first step to simplifying is to remember an important trig identity.

If we rewrite it to look like the denominator, it is.

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

The supplementary angle identity states that in the positive quadrants, if two angles are supplementary (add to 180 degrees), they must have the same sine. In other words, 

if , then 

Using this, we can see that

Thus, if , then  also.

Recall that the cosine and sine of complementary angles are equal.

Thus, we are looking for the complement of 70, which gives us 20.

When we take the cosine of this, we will get an equivalent statement. 

The cosine graph at zero has a peak at 1.

The first peak in the sine curve is at π/2, so you adjust the cosine with a -π/2 and that gives the answer 

Simplifying this expression relies on an understanding of the cofunction identities.  The cofunction identities hinge on the fact that the value of a trig function of a particular angle is equal to the value of the cofunction of the the complement of that angle.  In other words,

That means that returning to our initial expression, we can do some substiution.

We then turn to another identity, namely the fact that tangent is just the quotient of sine and cosine.

We can substitute again

Yet dividing by a fraction is the same as multipying by the reciprocal.

With some cancellation, we have arrived at our answer.

Think of a right triangle. The two non-right angles are complementary since the angles in a triangle add up to 180. As a result, when we apply the definitions of sine and cosine, we get that the sine of one of these angles is equivalent to the cosine of the other. As a result, the sine of one angle is equal to the cosine of its complement and vice versa. 

Therefore, when we look at the angles of a right triangle that includes a 60 degree angle we get 

.

Thus the 

Complementary angles are equal to one's  to others