Verified answer

1. 1- sinx = cosx

square both sides.

1 - 2sinx + sin^2 (x) = cos^2 (x)

or

1 - 2sinx = cos^2(x) - sin^2(x)

since cos^2(x) = 1 - sin^2(x), subsitute this in

to give

1 - 2sinx = 1 - 2sin^2(x)

or

sinx = sin^2(x).

A number equals it's square only when it = zero.

ie only if sinx = 0

ie at x = 0

#2

2sinx+cotx-cscx=0

ie

2sinx + (cosx/sinx) + (1/sinx) = 0

put everything over sinx

this gives

(2sin^2 (x) + cosx + 1 ) / sinx = 0

assuming sinx is not zero, and

multiplying by sinx will eliminate it from both sides.

so we have

2sin^2(x) + cosx + 1 = 0

but sin^2(x) = 1 - cos^2(x), substitute this in to give

2 - 2cos^2(x) + cosx + 1 = 0

or

-2cos^2(x) + cosx + 3 = 0

but, if we let y = cosx, we have a quardtic

in y ( a key trick here)

-2y^2 + y + 3 = 0

Multiply both sides by -1 to give:

2y^2 - y - 3 = 0

Use the quadratic formulat for roots, or notice that

this factors as

(y +1) (2y - 3) = 0

so y = -1 or y = -3/2

Now y = cosx, so y = -3/2 is impossible.

for cosx = -1 we know that

cos ((3/2)pi) = -1

#3

sin2x-4sinx-cosx+2=0

First, use the formula sin2x = 2sinx cosx

to give:

2cosx sinx -4sinx -cosx+ 2 = 0

Notice that this factors as:

(2sinx - 1) (cosx - 2)

so sinx = 1/2 or

cosx = 2 (impossible)

but

sin(pi/6) = 1/2

remedy right here systems of equations for x and y a million). y = 3 + 3x y = 18 - 2x 3x - y = -3 2x + y = 18 5x = 15 x = 3 y = 12 2). 18x + 6y = a hundred and twenty 4x + 3y = 40 8x + 6y = 80 10x = 40 x = 4 y = 8

#2 why did you change -cscx to + (1/sinx) at beginning ?