How can I do the following questions?

  1. How do you find the value of tan(67.5˚)

  2. How do you prove that costheta/(1 + sin theta) - costheta/(1+sintheta) = -2tantheta?

2 Answers
Jun 13, 2016

Start by putting the left side on a common denominator.

Explanation:

(costheta)/(1 + sin theta) - costheta/(1 - sin theta) = -2tantheta

(costheta(1 - sin theta))/((1 + sin theta)(1 - sin theta)) - (costheta(1 + sin theta))/((1 + sin theta )(1 - sin theta)) =

(costheta - costhetasintheta - costheta- costhetasintheta)/((1 + sin theta)(1 - sin theta)) =

(-2costhetasintheta)/((1 + sin theta)(1 - sin theta))=

(-2costhetasintheta)/(1 - sin^2theta)=

Use the pythagorean identity cos^2theta = 1- sin^2theta.

(-2costhetasintheta)/(cos^2theta)=

(-2sintheta)/costheta =

Now apply the quotient identity sintheta/costheta = tantheta.

-2tantheta =

Identity proved!

Hopefully this helps!

Jun 14, 2016

Solving number 30...

Explanation:

Note that 67 1/2˚ is exactly half of 135˚. This is especially relevant because we know the value of tan(135˚), since 135˚ has a reference angle of 45˚ and is located in the second quadrant, thus tan(135˚)=-1.

Looking at the given identity:

tan(2theta)=(2tan(theta))/(1-tan^2(theta))

If we let theta=67 1/2˚, then we see that

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 tan(135˚):

-1=(2tan(67 1/2˚))/(1-tan^2(67 1/2˚))

This will be easier to look at if we let u=tan(67 1/2˚):

-1=(2u)/(1-u^2)

Cross-multiply. We want to solve for u, since u=tan(67 1/2˚).

-1(1-u^2)=2u

u^2-1=2u

Solve like you normally would a quadratic equation (set it equal to 0):

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=(u-1)^2. Add 1 to both sides:

u^2-2u+1=1+1

(u-1)^2=2

Take the square root of both sides:

u-1=+-sqrt2

u=1+-sqrt2

Since u=tan(67 1/2˚):

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 67 1/2˚ is in the first quadrant, we know its tangent must be positive. Thus, we take the only positive solution out of the two:

tan(67 1/2˚)=1+sqrt2