Question

If A ,B and C are the interior angles of ∆ABC,show that tan A+B/2=cot c/2

Answer 1

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies A,B \: and \:C \: are \: interior \: angles \: of \:\triangle ABC \\ \\ \red{\underline \bold{To \: Show :}} \\ \tt: \implies tan \bigg(\frac{a + b}{2} \bigg) = cot\bigg(\frac{c}{2} \bigg)$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies \angle A + \angle B+ \angle C= 180 \degree \\ \\ \tt: \implies \angle A+ \angle B = 180 \degree - \angle C \\ \\ \text{Dividing \: both \: side \: by \: 2}\\ \\ \tt: \implies \frac{ \angle A + \angle B}{2} = \frac{180 \degree - \angle C}{2} \\ \\ \tt: \implies \frac{ \angle A+ \angle B}{2} = \frac{180 \degree}{2} - \frac{ \angle C}{2} \\ \\ \tt: \implies \frac{ \angle A+ \angle B }{2} =90 \degree - \frac{ \angle C}{2} \\ \\ \text{Taking \: trignometric \: function \: tan \: both \: side}\\ \\ \tt : \implies tan \bigg(\frac{ A+ B }{2} \bigg) =tan \bigg(90 \degree - \frac{C}{2} \bigg) \\ \\ \tt \circ \: tan( 90 \degree - \theta) = cot \: \theta\\ \\ \green{ \tt : \implies tan \bigg(\frac{A+ B}{2} \bigg) = cot \bigg(\frac{C}{2} \bigg ) }\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \green{ \huge{ \boxed{ \tt Proved}}}$

Answer 2

Answer:

⭐ QUESTION ⭐

✏ If A ,B and C are the interior angles of ∆ABC,show that tan A+B/2=cot c/2

⭐ GIVEN ⭐

✏ A, B and C are interior angle of ∆ABC

⭐ TO FIND ⭐

✏ tan(a+b/2) = cot(c/2)

⭐ ATQ ⭐

➡As we know the formula,

✳ ∠A + ∠B + ∠C = 180°

=> ∠A + ∠B = 180° - ∠C

✍ We have to divide both sides by 2, we get

=> ∠A + ∠B / 2 = 180° - ∠C / 2

=> ∠A + ∠B / 2 = 180°/2 - ∠C/2

=> ∠A + ∠B / 2 = 90° - ∠C/2

✍ We have to take tan both the sides, we get

=> tan(∠A +∠B / 2)=tan(90° -∠C/2

✍ We know the formula,

✳ tan(90°- θ) = cotθ

=> tan(∠A + ∠B / 2) = cot(c/2)

PROVED ✔

_________________

➡ Some important formula:-

1) sin(90° - θ) = cosθ

2) cos(90° - θ) = sinθ

3) tan(90° - θ) = cotθ

4) cosec(90° - θ) = secθ

5) sec(90° - θ) = cosecθ

6) cot(90° - θ) = tanθ

Step-by-step explanation:

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