# Solving Trig Identity: Sin[1/2 sin^−1 (x)] = 1/2 X (sqr(1+x) –sqr(1-x))

• MHB
• DUC_123
In summary, the conversation discusses an identity involving sin and a clue to solve the problem. The identity is sin[1/2 sin^−1 (x) ]=1/2 * (sqr(1+x) –sqr(1-x)). The individual tried using "2y=sin^−1 (x)", f=sin(y), and "x=sin(2y)" but could not find a solution.
DUC_123
Good morning, may I possibly get a clue into solving this problem : check this identity: sin[1/2 sin^−1 (x) ]=1/2 X (sqr(1+x) –sqr(1-x)). I tried to say "2y=sin^−1 (x)", Then f=sin(y), then "x=sin(2y)", x=2sin(x) X cos(x), but I can't find further. Thank you very much.

DUC_123 said:
Good morning, may I possibly get a clue into solving this problem : check this identity: sin[1/2 sin^−1 (x) ]=1/2 X (sqr(1+x) –sqr(1-x)). I tried to say "2y=sin^−1 (x)", Then f=sin(y), then "x=sin(2y)", x=2sin(y) X cos(y), but I can't find further. Thank you very much.

Please, no x's or X's. If you must use something use *. Many people just use a space, such as in x = 2 sin(y) cos(y).

-Dan

## 1. What is a trig identity?

A trig identity is an equation that is true for all values of the variables involved. It is used to simplify and manipulate trigonometric expressions.

## 2. How do you solve a trig identity?

To solve a trig identity, you must use algebraic manipulations and trigonometric identities to transform the given expression into a simpler form that is easier to evaluate.

## 3. What is the inverse trigonometric function?

The inverse trigonometric function is the inverse of a trigonometric function. It takes a ratio of sides in a right triangle and returns the angle that would produce that ratio.

## 4. How do you solve for x in the given trig identity?

To solve for x in the given trig identity, you must first use the inverse trigonometric function to isolate the trigonometric expression. Then, use algebraic manipulations to simplify the remaining expression and solve for x.

## 5. Can you explain the steps for solving Sin[1/2 sin^−1 (x)] = 1/2 X (sqr(1+x) –sqr(1-x))?

First, use the inverse sine function to rewrite the left side of the equation as sin^−1(x/2). Then, use the double angle formula for sine to simplify the expression to sin^−1(x/2) = 1/2(sin^−1(x) + sin^−1(x)). Next, use the identity sin^−1(x) = x to simplify the expression to 1/2(x + x) = x. Finally, use the identity sqr(1-x) = 1-x to rewrite the right side of the equation as x(sqr(1+x) – 1+x). The final step is to distribute the x and simplify the expression to x = 1/2(sqr(1+x) – sqr(1-x)).

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