Consider the following two statements: Statement 1: In sin A = , then value of cot A is . Statement 2: cot A = sin A.cos A. Which of the following is valid? 1. Both the statements are true. 2. Both the statements are false. 3. Statement 1 is true, and Statement 2 is false. 4. Statement 1 is false, and Statement 2 is true.

Given, sin A = By formula, ⇒ sin2 A + cos2 A = 1 ⇒ + cos2 A = 1 ⇒ + cos2 A = 1 ⇒ cos2 A = ⇒ cos2 A = ⇒ cos A = . By formula, ⇒ cot A = . Thus, both the statements are false. Hence, option 2 is the correct option.

Mathematics

Consider the following two statements:
Statement 1: In sin A = $\dfrac{1}{2}$ , then value of cot A is $\dfrac{1}{\sqrt{3}}$ .
Statement 2: cot A = sin A.cos A.
Which of the following is valid?
Both the statements are true.
Both the statements are false.
Statement 1 is true, and Statement 2 is false.
Statement 1 is false, and Statement 2 is true.

Trigonometrical Ratios

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Answer

Given, sin A = $\dfrac{1}{2}$

By formula,

⇒ sin² A + cos² A = 1

⇒ $\Big(\dfrac{1}{2}\Big)^2$ + cos² A = 1

⇒ $\dfrac{1}{4}$ + cos² A = 1

⇒ cos² A = $1 - \dfrac{1}{4}$

⇒ cos² A = $\dfrac{3}{4}$

⇒ cos A = $\sqrt{\dfrac{3}{4}} = \dfrac{\sqrt{3}}{2}$ .

By formula,

⇒ cot A = $\dfrac{\text{cos A}}{\text{sin A}} = \dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{1}{2}} = \dfrac{2\sqrt{3}}{2} = \sqrt{3}$ .

Thus, both the statements are false.

Hence, option 2 is the correct option.

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