The value of which of the following is definitely zero?

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Solution

Q.
The value of which of the following is definitely zero

$(i) s i n 45 ° - c o s 45 ° (i i) t a n 90 ° - c o t 90 ° (i i i) s i n 60 ° - c o s 30 ° (i v) t a n 30 ° - c o t 30 °$

1 both (i) and (iv)

2 both (i) and (iii)

3 only (i)

4 only (iii)

Ans.
(2)

$(i) s i n 45 ° = \frac{1}{\sqrt{2}}, c o s 45 ° = \frac{1}{\sqrt{2}} \Rightarrow s i n 45 ° - c o s 45 ° = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0 (i i) t a n 90 ° = \infty, c o t 90 ° = 0 \Rightarrow t a n 90 ° - c o t 90 ° = not defined (iii) \sin 60 ° = \frac{\sqrt{3}}{2}, \cos 30 ° = \frac{\sqrt{3}}{2} \Rightarrow \sin 60 ° - \cos 30 ° = \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0 (iv) \tan 30 ° = \frac{1}{\sqrt{3}}, \cot 30 ° = \sqrt{3} \tan 30 ° - \cot 30 ° = \frac{1}{\sqrt{3}} - \sqrt{3} = \frac{1 - 3}{\sqrt{3}} = - \frac{2}{\sqrt{3}} \neq 0$

Hence, (i) and (iii) will definitely give zero.

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