If sin A = 45\dfrac{4}{5}54, the value of 1−cos (90° - A)1+cos (90° - A)\sqrt{\dfrac{1 - \text{cos (90° - A)}}{1 + \text{cos (90° - A)}}}1+cos (90° - A)1−cos (90° - A) is :
1
3
12\dfrac{1}{2}21
13\dfrac{1}{3}31
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Given,
sin A = 45\dfrac{4}{5}54
We need to find the value of:
1−cos (90° - A)1+cos (90° - A)\sqrt{\dfrac{1 - \text{cos (90° - A)}}{1 + \text{cos (90° - A)}}}1+cos (90° - A)1−cos (90° - A)
Solving,
⇒1−cos (90° - A)1+cos (90° - A)⇒1−sin A1+sin A⇒1−451+45⇒5−455+45⇒1595⇒1×55×9⇒19⇒13.\Rightarrow \sqrt{\dfrac{1 - \text{cos (90° - A)}}{1 + \text{cos (90° - A)}}} \\[1em] \Rightarrow \sqrt{\dfrac{1 - \text{sin A}}{1 + \text{sin A}}} \\[1em] \Rightarrow \sqrt{\dfrac{1 - \dfrac{4}{5}}{1 + \dfrac{4}{5}}} \\[1em] \Rightarrow \sqrt{\dfrac{\dfrac{5 - 4}{5}}{\dfrac{5 + 4}{5}}} \\[1em] \Rightarrow \sqrt{\dfrac{\dfrac{1}{5}}{\dfrac{9}{5}}} \\[1em] \Rightarrow \sqrt{\dfrac{1 \times 5}{5 \times 9}} \\[1em] \Rightarrow \sqrt{\dfrac{1}{9}} \\[1em] \Rightarrow \dfrac{1}{3}.⇒1+cos (90° - A)1−cos (90° - A)⇒1+sin A1−sin A⇒1+541−54⇒55+455−4⇒5951⇒5×91×5⇒91⇒31.
Hence, Option 4 is the correct option.
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