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We will learn how to prove the property of the inverse trigonometric function arccos(x) - arccos(y) = arccos(xy + β1βx2β1βy2)
Proof:
Let, cosβ1 x = Ξ± and cosβ1 y = Ξ²
From cosβ1 x = Ξ± we get,
x = cos Ξ±
and from cosβ1 y = Ξ² we get,
y = cos Ξ²
Now, cos (Ξ±
- Ξ²) = cos Ξ± cos Ξ² + sin Ξ± sin Ξ²
β cos (Ξ± - Ξ²) = cos Ξ± cos Ξ² + β1βcos2Ξ± β1βcos2Ξ²
β cos (Ξ± - Ξ²) = (xy + β1βx2β1βy2)
β Ξ± - Ξ² = cosβ1(xy + β1βx2β1βy2)
or, cosβ1 x - cosβ1 y = cosβ1(xy + β1βx2β1βy2)
Therefore, arccos(x) - arccos(y) = arccos(xy) + β1βx2β1βy2) Proved.
Note: If x > 0, y > 0 and x2 + y2 > 1, then the cosβ1 x + sinβ1 y may be an angle more than Ο/2 while cosβ1(xy - β1βx2β1βy2), is an angle between β Ο/2 and Ο/2.
Therefore, cosβ1 x - cosβ1 y = Ο - cosβ1(xy + β1βx2β1βy2)
β Inverse Trigonometric Functions
11 and 12 Grade Math
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