In an angle-side-angle problem, Law of Sines will solve the triangle. 

First find angle A:

Then use Law of Sines.

One leg is 4cm and the hypotenuse is 5cm. Plug in 4 for one of the legs and 5 for the hypotenuse (c).

Subtracting 16 from either side of the equation gives us:

The last step is to take the square root both sides resulting in:

Find a common denominator

Multiply the numerators and leave the denominators factored

Add numerators

Pythagorean identity

Combine like terms

Factor numerator

Reduce

Reciprocal identity

The easiest identity to use here is:

Substituting in the given values we get:

To solve this problem we need to rewrite it in terms of  and .

Rewriting cotangent we get the following.

Rewriting secant we get the following.

Thus, we can substitute these identities into our original problem and simplify.

We can work with either side of the equation as we choose. We work with the right hand side of the equation since there is an obvious double angle here. We can factor the numerator to receive the following:

Next we note the power reducing formula for sine so we can extract the necessary components as follows:

The power reducing formula must be inverted giving:

Now we can distribute and reduce:

Finally recalling the basic identity for the cotangent:

This proves the equivalence.

The power reducing formulas for both sine and cosine differ in only the operation in the numerator. Applying the power reducing formulas here we get:

Multiplying the binomials in the numerator and multiplying the denominators:

Reducing the numerator:

We again use the power reducing formula for cosine as follows:

Combining the numerator by determining a common denominator:

Now simply reducing the double fraction:

In this case we choose to work with the side that appears to be simpler, the left hand side. We begin by using the power reducing formulas:

Next we perform the multiplication on the numerator:

The next step we take is to remove the double angle, since there is no double angle in the alleged solution:

Finally we multiply the binomials in the numerator on the left hand side to determine if the equivalence holds:

We see that the equivalence does not hold.

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 .