Question

Trigonometry Question. Please Help fast.

ps. i'll report you if you don't answer properly.
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Answer 1

GIVEN :-

• (cot² 15° - 1)/(cot² 15° + 1)

TO FIND :-

• The value of (cot² 15° - 1)/(cot² 15° + 1)

SOLUTION :-

★ Firstly we will find the value of cot 15°.

⇒ cos 2A = 1 - sin² A

⇒ 1 - cos 30° = sin²15°

⇒ (2 - √3) / 2 = sin²15°

⇒ sin 15° = √{(2 - √3) / 2}

similarly,

⇒ cos 2A = 2cos² A- 1

⇒ cos 30° = 2cos²15° - 1

⇒ √3 / 2 = 2cos²15° - 1

⇒ cos² 15° = ( 2 + √3) / 2

⇒ cos 15° = √{(2 + √3) / 2}

Now,

⇒ cot A = cos A / sin A

⇒ cot 15° = cos 15° / sin15°

⇒ cot 15° = √{ (2 + √3) / 2 } / √{ ( 2 - √3) / 2 }

⇒ cot 15° = √{ (2 + √3) / (2 - √3) }

⇒ cot 15° = √{ (2 + √3)² / (2² - (√3)²) }

⇒ cot 15° = 2 + √3

Hence the required value of cot 15° is 2 + √3.

⇒ (cot² 15° - 1)/(cot² 15° + 1)

As we know that , cot 15° = 2 + √3.

⇒ [{(2 + √3)² - 1}/{(2 + √3)² + 1}]

By using identity :- (a + b)² = a² + b² + 2ab

⇒ [{(2)² + (√3)² + 2 × 2√3 - 1}/{(2)² + (√3)² + 2 × 2√3 + 1]

⇒ [(4 + √9 + 4√3 - 1)/(4 + √9 + 4√3 + 1)

⇒ [(4 + 3 + 4√3 - 1)/(4 + 3 + 4√3 + 1)]

⇒ [(7 - 1 + 4√3 )/8 + 4√3 ]

⇒ (6 + 4√3)/(8 + 4√3)

⇒ (3 + 2√3)/(4 + 2√3)

By rationalising the denominator we get,

⇒ {(3 + 2√3)(4 - 2√3)}/{(4 + 2√3)(4 - 2√3)}]

⇒ [{12 - 6√3 + 8√3 - 12 }/{(4)² - (2√3)²]

⇒ [{12 - 6√3 + 8√3}/{16 - 12}]

⇒ (-6√3 + 8√3)/4

⇒ 2√3/4

⇒ √3/2

Hence the required value of (cot² 15° - 1)/(cot² 15° + 1) is √3/2.

Answer 2

Given :

• $\\\implies\bf{ \frac{cot^2\:15°\:-\:1}{cot^2\:15°\:+\:1}}$

According to the question :

Dividing 15π by 180, we get,

$\\\implies\bf{ \frac{cot^2\:(π/12)\:-\:1}{cot^2\:(15π/180)\:+\:1}}$

$\\\implies\bf{cot{ \frac{π}{12}}}$

$\\\implies\bf{ \frac{cos\:(π/6)\:+\:1}{sin\:(π/6)}}$

$\\\implies\bf{ \sqrt{3}}\:+\:2$

$\\\implies\bf{ \frac{2\:+\:{ \sqrt{3}}\:-\:1}{cot^2\:(15π/180)\:+\:1}}$

Dividing 15π by 180, we get,

$\\\implies\bf{ \frac{2\:+\:{ \sqrt{3}}\:-\:1}{cot^2\:(π/12)\:+\:1}}$

$\\\implies\bf{cot{ \frac{π}{12}}}$

$\\\implies\bf{ \frac{cos\:(π/6)\:+\:1}{sin\:(π/6)}}$

$\\\implies\bf{ \sqrt{3}}\:+\:2$

Dividing,

$\\\implies\bf{ \frac{2\:+\:{ \sqrt{3}}^2\:-\:1}{2\:+\:{ \sqrt{3}}^2\:+\:1}}$

$\\\implies\bf{ \sqrt{3}}\:+\:2$

So Its Done !!