2tanx-sin2x÷2sin*2x =tanx
What about it?
are you solving for x ?
are you proving that it is an identity ?
LS = 2tanx-sin2x÷2sin*2x <---- assuming you mean: 2tanx - sin(2x)/(2sin^2 x)
= 2sinx/cosx - 2sinxcox/(2sin^2 x)
= 2 sinx/cosx - cosx/sinx
= (2sin^2 x - cos^2 x)/(sinxcosx)
= (2sin^2 x - 1 + cos^2 x) / sinxcosx
= (3sin^2 x -1)/sinxcosx
≠ tanx
try solving:
(3sin^2 x -1)/sinxcosx = tanx = sinx/cosx
3sin^2 x - 1 = sin^2 x
2 sin^2 x = 1
sinx = ± 1/√2
x = 45° or 135° OR x = π/4, 3π/4 in radians
Having tanx on both sides seems redundant.
Maybe you meant
(2tanx - sin2x) / (2sin^2x)
= (2sinx/cosx - 2sinx cosx) / (2sin^2x)
= (1/cosx - cosx)/sinx
= (1 - cos^2x)/(sinx cosx)
= sin^2x / (sinx cosx)
= sinx/cosx
= tanx
To simplify the given expression and prove that it is equal to tanx, let's break down each term step by step:
1. Start with the left side of the equation: 2tanx - sin2x ÷ 2sin*2x.
2. The first term, 2tanx, can be simplified using the trigonometric identity: tanx = sinx/cosx.
Substitute tanx with sinx/cosx: 2(sin(x)/cos(x))
Simplify by multiplying 2 to the numerator: 2sinx/cosx
3. The second term, sin2x, can be further simplified as sin(2x) using the double angle formula: sin(2x) = 2sinx*cosx.
Substitute sin2x with 2sinx*cosx: 2sinx*cosx ÷ 2sin*2x
Simplify by canceling out the 2 terms: sinx*cosx ÷ sin*2x
4. The third term, 2sin*2x, can be rewritten as sin(2x).
Substitute sin*2x with sin(2x): sinx*cosx ÷ sin(2x)
5. Now, the expression becomes: 2sinx/cosx - sinx*cosx ÷ sin(2x)
6. To simplify further, consider that sin(x) / cos(x) is equal to tan(x).
Substitute 2sinx/cosx with tanx: tanx - sinx*cosx ÷ sin(2x)
7. We can express sin(2x) as 2sinxcosx using the double angle formula.
Substitute sin(2x) with 2sinxcosx: tanx - sinx*cosx ÷ 2sinxcosx
8. Simplify by canceling out the sinx term in the numerator and denominator:
tanx - cosx ÷ 2cosx
9. Finally, divide each term in the numerator by cosx:
tanx - 1/2
10. The final simplified expression is: tanx - 1/2 = tanx.
Therefore, we have successfully simplified the left side of the equation to be equal to tanx.