Question

Simplify

Answer 1

@FaiqRaees

Answer 2

Use the product of a sum and a difference in the numerator and the denominator.
Then use identities.

Answer 3

?

Answer 4

What is (a + b)(a - b) equal to?

Answer 5

wouldn't it just be sin

Answer 6

because the top is 0

Answer 7

Top is not 0.

Answer 8

Notice the top and bottom both follow the pattern (a + b)(a - b) correct?

Answer 9

The product of a sum and a difference is the difference of two squares.

\((a + b)(a - b) = a^2 - b^2\)

Answer 10

so sin^2-sin^2?

Answer 11

No.
Look in the numerator first.

\((1 - \cos \theta)(1 + \cos \theta)\)

Here, a is 1 and b is cos theta

so you get:

\((1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta\)

ok?

Answer 12

yes

Answer 13

Now do the same with the denominator. It follows the same pattern only with sin instead of cos.

Answer 14

so 1-sin^2theta

Answer 15

\(\dfrac{(1 - \cos \theta)(1 + \cos \theta)}{(1 - \sin \theta)(1 + \sin \theta)} =\)

\(= \dfrac{1 - \cos^2 \theta}{1 - \sin^2 \theta} \)

Good. So far we are here.

Now use the same trig identity in the numerator and denominator to end up with a single trig function in each.

Answer 16

wouldn't the 1 just cancel

Answer 17

and the neg

Answer 18

Trig identity:

\(\sin^2 \theta + \cos^2 \theta = 1\)

Use this form for the numerator:

\(\sin^2 \theta =1 - \cos^2 \theta\)

Use this form for the denominator:

\(\cos^2 \theta =1 - \sin^2 \theta\)

Answer 19

No. You can't cancel a part of a sum or difference. You can only cancel entire factors.

Answer 20

Ok idk then