Simplify (sin^2)*(67.5deg) - (cos^2)*(67.5deg) and then evaluate exactly.
well, cos 2x = cos^2 x - sin^2 x, so what do you think?
If you end up with 1/√2 you are correct.
How exactly do you get there though? I tried to follow a similar homework problem and got:
= (cos^2)*(67.5deg) - (sin^2)*(67.5deg)
= cos(2*67.5deg)
= cos135deg
= -√2/2
note that the problem was
sin^2(67.5)-cos^2(67.5)
= -cos(135)
my hint was not intended to make you forget what was originally asked. :-)
And, I know lots of folks hate roots in the denominator, but I see no reason why √2/2 is better than 1/√2. Just mt preference
Thank you very much!
To simplify the expression (sin^2)(67.5°) - (cos^2)(67.5°) and evaluate it exactly, we can use the trigonometric identity:
cos^2(θ) = 1 - sin^2(θ)
Using this identity, we can rewrite the expression as:
(sin^2)(67.5°) - (1 - sin^2)(67.5°)
Let's evaluate each part step by step:
First, let's find sin(67.5°):
We can use a half-angle formula for sine to find sin(67.5°):
sin(67.5°) = sqrt[(1 - cos(135°))/2]
Now, let's find cos(135°):
cos(135°) is a special angle that we can find by knowing that it lies in the 2nd quadrant, where cos is negative. Thus, cos(135°) = -sqrt(2)/2.
Now, substituting this value back into the equation:
sin(67.5°) = sqrt[(1 - (-sqrt(2)/2))/2]
= sqrt[(1 + sqrt(2))/2]
= (1 + sqrt(2))/2
Now, let's calculate sin^2(67.5°):
sin^2(67.5°) = [(1 + sqrt(2))/2]^2
= (1 + 2sqrt(2) + 2)/4
= (3 + 2sqrt(2))/4
Next, let's simplify cos^2(67.5°):
cos^2(67.5°) = 1 - sin^2(67.5°)
= 1 - (3 + 2sqrt(2))/4
= (4 - 3 - 2sqrt(2))/4
= (1 - 2sqrt(2))/4
Finally, substitute back into the original expression and simplify:
(sin^2)(67.5°) - (cos^2)(67.5°)
= (3 + 2sqrt(2))/4 - (1 - 2sqrt(2))/4
= (3 + 2sqrt(2) - 1 + 2sqrt(2))/4
= (2 + 4sqrt(2))/4
= 1/2 + sqrt(2)
Therefore, the simplified and exact value of the expression (sin^2)(67.5°) - (cos^2)(67.5°) is 1/2 + sqrt(2).