Find the value of one minus tan squared of seven 𝜋 over eight divided by one plus tan squared of seven 𝜋 over eight without using a calculator.
Okay, as we start to evaluate this expression, the first thing we can notice is that both arguments of the tangent function are seven 𝜋 over eight. As a bit of shorthand going forward, let’s represent this fraction by 𝛼. This means our expression can be written one minus the tan squared of 𝛼 divided by one plus the tan squared of 𝛼.
At this point, we can recall that the tangent of a given angle is equal to the sine of that angle divided by the cosine of that angle. And this implies, squaring both sides, that the tan squared of 𝑥 equals the sin squared of 𝑥 over the cos squared of 𝑥. In the expression we want to evaluate then, we can replace tan squared of 𝛼 with sin squared 𝛼 over cos squared 𝛼. And then we can multiply both numerator and denominator of this fraction by the cos squared of 𝛼. That gives us this result. And as it turns out, we can simplify both the numerator and denominator of this fraction using identities.
Regarding the denominator, the Pythagorean identity says that the sine squared of an angle plus the cosine squared of that same angle equals one. This means our entire denominator is one. And for our numerator, we can recall one of the double angle identities for the cosine function. This tells us that if we have the cosine squared of an angle minus the sine squared of that same angle, it’s equal to the cosine of two times that angle. This whole fraction we want to evaluate then simplifies to the cos of two 𝛼 over one or simply the cos of two 𝛼.
What we found then is that our original expression equals the cos of two 𝛼, where 𝛼 is seven 𝜋 over eight. So two times 𝛼 is seven 𝜋 over four. We want to evaluate this expression without a calculator. And to do that, we can consider the unit circle. On this circle, an angle of seven 𝜋 over four would look like this, where the angle to complete one full revolution, so to speak, is 𝜋 over four.
Along with this, we can recall that, in both the first and the fourth quadrants of this circle, the cosine function is positive. This fact, along with the fact that our angle of seven 𝜋 over four perfectly bisects or divides in half this fourth quadrant, means that we can equivalently calculate the cos of 𝜋 over four. And we’ll get the same answer. 𝜋 over four radians is the same as 45 degrees. And the cosine of this angle equals exactly one over the square root of two. That’s our final answer. But note that we could equivalently say this is the square root of two over two.