Which of the following would be an acceptable first step in simplifying the expression sinx/1 + sinx.
1/(cscx + sinx)
sinx + 1
(sinx(1 - sinx))/((1 + sinx)(1 - sinx))

Solution:

Given, the expression is sinx/(1 + sinx).

We have to find the first step in simplifying the expression.

By taking conjugate,

= \(\frac{sinx}{1+sinx}\times \frac{1-sinx}{1-sinx}\)

By using algebraic identity,

(a - b)(a + b) = (a² - b²)

= \(\frac{sinx(1-sinx)}{(1+sinx)(1-sinx)}\\=\frac{sinx(1-sinx)}{1-sin^{2}x}\)

By using trigonometric identity,

sin²x + cos²x = 1

cos²x = 1 - sin²x

So, \(\frac{sinx(1-sinx)}{1-sin^{2}x}=\frac{sinx(1-sinx)}{cos^{2}x}\)

\(=\frac{sinx-sin^{2}x}{cos^{2}x}\\=\frac{sinx}{cos^{2}x}-\frac{sin^{2}x}{cos^{2}x}\\=tanxsecx-tan^{2}x\)

Taking out common term,

= tanx[secx - tanx]

Therefore, the first step is taking conjugate \(\frac{sinx}{1+sinx}\times \frac{1-sinx}{1-sinx}\).

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Which of the following would be an acceptable first step in simplifying the expression sinx/1+sinx.

Summary:

\(\frac{sinx}{1+sinx}\times \frac{1-sinx}{1-sinx}\) would be an acceptable first step in simplifying the expression sinx/1 + sinx.