trig prob 2

Question

1. Simplify: (CotA+1)/tanA+1 3. Prove that: [(Sin2A-1) / (sin A - cos A)] = cos A - sin A 4. Calculate the value of sin - 75 degrees, without a calculator. 5. Derive a formula for cos 2A in terms of sin A.

Answer 1

1)

(cotA+1)/tanA+1

cotA can be written as 1/ tan A.

Answer 2

4)

sin (-75°) = - sin 75°

                = - sin (45 + 30 )

Using the sum formula : sin (A + B ) = sin A cos B + cos A sinB

= - [ sin 45 cos 30 + cos 45 sin 30]

= - [ (1/√2)(√3/2) + (1/√2)(1/2)]

= - [√3/ 2√2 + 1/2√2]

= - [(√3 +1 )/ 2√2]

Answer 3

5)

cos2A  = cos ( A + A )

           = cos A cos A - sin A sin A              (Using trigonometric sum formula:

                                                                            cos (A + B) = cos A cos B - sin A sin B)

          = cos² A - sin² A 

         = (1 - sin² A) - sin² A 

          = 1 - 2 sin² A

Answer 4

3.

Left hand identity:

[sin (2A) - 1] / (sin A - cos A)

= [2sinA cos A - (sin² A + cos² A)]/ (sin A - cos A)

= - [ (sin² A + cos² A) - 2sinA cos A]/ (sin A - cos A)

= - (sin A - cos A )²/ (sin A - cos A)

= - (sin A - cos A )

= -sin A + cos A

= Left hand identity