How can I do the following questions?
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How do you find the value of tan(67.5˚)
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How do you prove that costheta/(1 + sin theta) - costheta/(1+sintheta) = -2tantheta ?
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How do you find the value of
tan(67.5˚) -
How do you prove that
costheta/(1 + sin theta) - costheta/(1+sintheta) = -2tantheta ?
2 Answers
Start by putting the left side on a common denominator.
Explanation:
Use the pythagorean identity
Now apply the quotient identity
Identity proved!
Hopefully this helps!
Solving number 30...
Explanation:
Note that
Looking at the given identity:
tan(2theta)=(2tan(theta))/(1-tan^2(theta))
If we let
tan(2xx67 1/2˚)=(2tan(67 1/2˚))/(1-tan^2(67 1/2˚))
tan(135˚)=(2tan(67 1/2˚))/(1-tan^2(67 1/2˚))
We already know the value of
-1=(2tan(67 1/2˚))/(1-tan^2(67 1/2˚))
This will be easier to look at if we let
-1=(2u)/(1-u^2)
Cross-multiply. We want to solve for
-1(1-u^2)=2u
u^2-1=2u
Solve like you normally would a quadratic equation (set it equal to
u^2-2u-1=0
You could use the quadratic formula here, but I'll complete the square:
u^2-2u=1
We want the left side to match
u^2-2u+1=1+1
(u-1)^2=2
Take the square root of both sides:
u-1=+-sqrt2
u=1+-sqrt2
Since
tan(67 1/2˚)=1+-sqrt2
However, something's up... what should we do about the plus or minus sign? The tangent of a single angle can only equal one thing.
Since
tan(67 1/2˚)=1+sqrt2