How can we answer this question?
1. sin2(x) - 2 = -sin(x)
This can't be an identity because it is not always true. For instance...
sin2(0) - 2 = -sin(0)
02 - 2 = -0
-2 = 0 This is false, so this equation is not an identity.
However, this equation does have solutions. We can find its solutions.
sin2(x) - 2 = -sin(x)
Add sin(x) to both sides of the equation.
sin2(x) + sin(x) - 2 = 0
Factor the left side like this: u2 + u - 2 = (u + 2)(u - 1)
(sin(x) + 2)(sin(x) - 1) = 0
Set each factor equal to zero.
sin(x) + 2 = 0 or sin(x) - 1 = 0
sin(x) = -2 or sin(x) = 1
sin returns values between -1 and 1 inclusively, so we can rule out the first option. That leaves...
sin(x) = 1
x = arcsin(1) = 90°
There will also be a solution every time 360° is added to 90° , so all the real solutions are
x = 90° + 360°n where n is an integer
Does this answer your question? I might have misunderstood your question.
2. (tan2x) / (1 + sec x) = sec x + 1 Let's test whether this is true when x = 0 .
(tan2x) / (1 + 1/cosx) = 1/cosx + 1
(tan20) / (1 + 1/cos0) = 1/cos0 + 1
(02) / (1 + 1/1) = 1/1 + 1
0 = 2 This is false, so this equation is not true for all values of x and so it is not an indentity.
If you want to find the solutions to the equation, we can try to find them.
(tan2x) / (1 + sec x) = sec x + 1 Multiply both sides by (1 + sec x)
tan2x = (sec x + 1)(1 + sec x)
tan2x = sec x + sec2x + 1 + sec x
tan2x = sec2x + 2sec x + 1 Rewrite tan and sec in terms of sin and cos.
sin2x / cos2x = 1/cos2x + 2/cos x + 1 Multiply through by cos2x
sin2x = 1 + 2cos x + cos2x Subtract sin2x from both sides.
0 = 1 + 2cos x + cos2x - sin2x Substitute 1 - cos2x in for sin2x
0 = 1 + 2cos x + cos2x - (1 - cos2x)
0 = 1 + 2cos x + cos2x - 1 + cos2x
0 = 2cos x + 2cos2x Divide through by 2 .
0 = cos x + cos2x Factor cos x out of both terms.
0 = cos x(1 + cos x) Set each factor equal to zero.
cos x = 0 or 1 + cos x = 0
However, neither of these can be true.
If cos x = 0 then sec x is undefined and the original equation is undefined.
If cos x = -1 then 1 + sec x = 1 - 1 = 0 and the original equation is undefined.
So this equation has no solutions.