Question

Use an identity to find the exact value of the expression:

Note: You are not allowed to use decimals in your answer.

tan(67.5∘)) =

Answer 1

Explanation:

We can use the half-angle identity to find the exact value of tan(67.5°). The half-angle identity for tangent is:

tanleft(fractheta2right)=pmsqrtfrac1−cos(theta)1+cos(theta)

Since 67.5° is half of 135°, we can use the identity to find the value of tan(67.5°):

tan(67.5°)=tanleft(frac135°2right)=pmsqrtfrac1−cos(135°)1+cos(135°)

We know that cos(135°) = -√2/2, so we can substitute this value into the identity:

tan(67.5°)=pmsqrtfrac1−(−√2/2)1+(−√2/2)=pmsqrtfrac1+√2/21−√2/2

We can rationalize the denominator by multiplying both the numerator and denominator by (1+√2/2):

tan(67.5°)=pmsqrtfrac(1+√2/2)(1+√2/2)(1−√2/2)(1+√2/2)=pmsqrtfrac(1+√2/2)21−(√2/2)2=pmsqrtfrac(1+√2/2)21−1/2=pm(1+√2/2)

Since 67.5° is in the first quadrant, where tangent is positive, we take the positive root:

tan(67.5°)=1+frac√22

So, the exact value of tan(67.5°) is 1 + √2/2.