# CAT Practice : Trigonometry

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Max and Min value of the given expression

## Trigonometric Identities

Q.14: If $\frac{(2 Sin x) }{(1+cos⁡ x + Sinx)} = t, \frac{(1 – Cos x + Sin x) }{(1 + Sin x)}$ can be written as:
1. $\frac{1}{t}$
2. t
3. √t
4. $\frac{t}{Sin x}$

Choice B. t

## Detailed Solution

= $\frac{(2 Sin x) }{(1+cos⁡ x + Sinx)} * \frac{(1+Sin⁡ x - Cos x) }{(1+Sin⁡x - Cos x)}$

= 2 Sin x * $\frac{(1 + Sin x – Cos x)}{((1 + Sin x)^2 – Cos^2 x)}$

= 2 Sin x * $\frac{(1 + Sin x – Cos x)}{(1 + Sin^2 x + 2 Sin x – Cos^2 x)}$

= 2 Sin x . $\frac{(1 + Sin x – Cos x)}{(Cos^2 x + Sin^2 x + Sin^2 x + 2 Sin x – Cos^2 x)}$

(Since, 1 = Cos2x + Sin 2x)

= 2 Sin x * $\frac{(1 + Sin x – Cos x)}{(2 Sin 2x + 2 Sin x)}$

= 2 Sin x * $\frac{(1 + Sin x – Cos x)}{(2 Sin x (1 + Sin x))}$

= $\frac{(1 – Cos x + Sin x)}{(1 + Sin x)}$ = t

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