Will somebody please help me with these Pre Calculus questions?

Question

The expression tanX+1/1-tanX can be rewritten as which of the following?

A. tan (x - (5π/4))

B. tan (x + (3π/4))

C. tan (x + (7π/4))

D. tan (x - (7π/4))

What is the exact value of sin(195°)?

A. (√2 + √6)/4

B. (√6 - √2)/4

C. (√2 - √6)/4

D. (√6 - √3)/4

What are all the solutions to the equation 1 - 2sin² θ = √3​/2

A. x = (π/6) + 2πn, x = (11π/6)+ 2πn

B. x = (π/8) + πn, x = (7π/8)+ πn

C. x = (π/4) + 2πn, x = (7π/4)+ 2πn

D. x = (π/12) + πn, x = (11π/12)+ πn

Answer 1

1-sin^2(x) = cos^2(x) = sqr3/2

cosx = sqr((sqr3)/2)

cosx = 21.4707 degrees= about .12pi = 12pi/100 = 3pi/25

3pi/25 is closest to B) among the 4 choices

try B) x=pi/8= 45/2= 22.5 degrees

1-sin^2(22.5) = .854 which is closer to sqr3/2 = 1.732/2 = .866

.854 is = about .866

x=24 degrees works even better = 24pi/180 6pi/45

1-sin^2(x) = sqr3/2

cos^2(x) = sqr3/2

cosx = sqr(sqr3/2)

x = cos^-1(sqr(sqr3/2))

x =about .1192816746pi radians

= about .12pi = 12pi/100 = 3pi/25 which is closest to pi/8 and answer B)

2nd problem, sine of 195 degrees

sin195 = sin(180+15) = sin180cos15 + cos180(sin15)

simplify

or just

use a calculator with trig and square root functions:

sin15 = about .2588

2nd choice: (sqr6-sqr2)/4 = about .2588

it's the correct answer

1st problem

(tanx +1)/(1-tanx)

= tan(x+7pi/4)

the 3rd choice

Answer 2

For question 1), use the tangent angle sum identity:

If you select an angle β where tan(β) = 1, you will end up with (tan(α) + 1)/(1 - tan(α)) which will match your expression. So, go through each answer (A - D) and find the one that where where tan(β) = 1. Example, on answer A, tan(-5π/4) = -1 so that is NOT going to work. etc.

For question 2), use the angle sum identity for sine:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

where x = 45° and y = 150°

For question C), use the double angle identity cos(2x) = 1 - sin²(x)