Given below are two statements: Statement I: ( 30^()+1)/( 30^()-1)=2( 30^()+1) Statement II : 2 45^() 45^()- 45^() 45^()=0 Option 1: Both Statement I and Statement II are true Option 2: Both Statement I and Statement II are false Option 3: Statement I is true but Statement II is false Option 4: Statement I is false but Statement II is true

Answer: Both Statement I and Statement II are true. Solution: Statement I: ( 30° + 1)/( 30° - 1) = 2( 30° + 1) 1) Left side: - 30° = √3/1 - So (√3 + 1)/(√3 - 1) - Rationalize denominator: ((√3 + 1)(√3 + 1))/((√3 - 1)(√3 + 1)) = (3 + 2√3 + 1)/(3 - 1) = (4 + 2√3)/2 = 2 + √3 2) Right side: 30° = √3/2 2(√3/2 + 1) = √3 + 2 = 2 + √3 Therefore Statement I is TRUE. Statement II: 2 45° 45° - 45° 45° = 0 1) 45° = 45° = 1/√2 2) 45° = 45° = 1 3) Let's substitute: 2(1/√2)(1/√2) - (1)(1) = 2(1/2) - 1 = 1 - 1 = 0 Therefore Statement II is TRUE. Hence, both statements are true.

Geometry

Trigonometry

Medium

Given below are two statements:

Statement I: $\frac{\cot 30^{\circ}+1}{\cot 30^{\circ}-1}=2\left(\cos 30^{\circ}+1\right)$

Statement II : $2 \sin 45^{\circ} \cos 45^{\circ}-\tan 45^{\circ} \cot 45^{\circ}=0$

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is true but Statement II is false

Statement I is false but Statement II is true

✅ Correct Option: 1

Statement I: $\frac{\cot 30° + 1}{\cot 30° - 1} = 2(\cos 30° + 1)$

Left side:

$\cot 30° = \frac{\sqrt{3}}{1}$ $cot 30° = \frac{3}{1}$
- So $\frac{\sqrt{3} + 1}{\sqrt{3} - 1}$
- Rationalize denominator: $\frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}$

= $\frac{3 + 2\sqrt{3} + 1}{3 - 1}$

= $\frac{4 + 2\sqrt{3}}{2}$

= $2 + \sqrt{3}$

Right side:

$\cos 30° = \frac{\sqrt{3}}{2}$

$2(\frac{\sqrt{3}}{2} + 1)$

= $\sqrt{3} + 2$

= $2 + \sqrt{3}$

Therefore Statement I is TRUE.

Statement II: $2\sin 45° \cos 45° - \tan 45° \cot 45° = 0$

$\sin 45° = \cos 45° = \frac{1}{\sqrt{2}}$
$\tan 45° = \cot 45° = 1$

Let's substitute:

$2(\frac{1}{\sqrt{2}})(\frac{1}{\sqrt{2}}) - (1)(1)$

= $2(\frac{1}{2}) - 1$

= $1 - 1$

= $0$

Therefore Statement II is TRUE.

Hence, both statements are true.

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Statement II: if $\cot (\theta)=\frac{12}{5}$ , then the value of $\sin (\theta)=\frac{5}{12}$

In the light of the above statements, choose the correct answer form the question below:

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