sec(sq) X - Sec(sq) X * Sin (sq) X = 1
To solve this equation, let's simplify it step by step.
1. Use the trigonometric identity: sec^2(x) = 1 + tan^2(x).
2. Replace sec^2(x) with 1 + tan^2(x) in the equation.
(1 + tan^2(x)) * x - (1 + tan^2(x)) * sin^2(x) = 1
3. Distribute the x and simplify:
x + tan^2(x)*x - sin^2(x)*x - tan^2(x)*sin^2(x) = 1
4. Rearrange the terms:
(1 - sin^2(x))*x + (tan^2(x) - tan^2(x)*sin^2(x)) = 1
5. Simplify the expression in the parentheses:
cos^2(x)*x + tan^2(x)*(1 - sin^2(x)) = 1
6. Use the trigonometric identity: 1 - sin^2(x) = cos^2(x).
cos^2(x)*x + tan^2(x)*cos^2(x) = 1
7. Factor out cos^2(x):
cos^2(x) * (x + tan^2(x)) = 1
8. Divide both sides by cos^2(x):
x + tan^2(x) = 1/cos^2(x)
9. Since tan(x) = sin(x)/cos(x), substitute tan^2(x) with sin^2(x)/cos^2(x):
x + sin^2(x)/cos^2(x) = 1/cos^2(x)
10. Multiply both sides by cos^2(x) to get rid of the denominator:
x*cos^2(x) + sin^2(x) = 1
11. Use the trigonometric identity: sin^2(x) = 1 - cos^2(x).
x*cos^2(x) + (1 - cos^2(x)) = 1
12. Simplify the expression in the parentheses:
x*cos^2(x) + 1 - cos^2(x) = 1
13. Combine like terms:
x*cos^2(x) - cos^2(x) + 1 = 1
14. Group the terms with cos^2(x):
(x - 1)*cos^2(x) + 1 = 1
15. Subtract 1 from both sides:
(x - 1)*cos^2(x) = 0
16. Since the product of two factors is zero, either (x - 1) = 0 or cos^2(x) = 0.
17. Solve for x:
(x - 1) = 0 ---> x = 1
18. Now, let's solve cos^2(x) = 0. Since cosine squared cannot be zero, there are no solutions for this case.
Therefore, the only solution to the equation is x = 1.