Question

Answer this..... prove using multiple angle formula........... ​

Answer 1

$\mathcal{\huge{\underline{\underline {\red{Question:-}}}}}$

✒ $cot \: \frac{a}{2} - tan \frac{a}{2} = 2cot \: a$

$\mathcal{\huge{\underline{\underline {\pink{Proof:-}}}}}$

cotA/2-tanA/2

=>(cosA/2 ÷ sin A/2) - (sinA/2 ÷ CosA/2)

=> (cos²A/2-sin²A/2)/sinA/2*cosA/2

⭐ We, know that cos²x-sin²x=cos2x

=> cos2(A/2)/ sinA/2*cosA/2

Multiplying and dividing by 2

=>cosA / sinA/2 ×cosA/2 × 2/2

=>2cosA/2 sinA/2×cosA/2

⭐ We know that sin2x=sinx × cosx

So,

=> 2cosA/sinA {cosa/sina=cota}

=> 2cotA

Hence, cotA/2 - tanA/2 = 2cotA

Proved ✔✔✔✔

____________________________

Answer 2

Question

Show that:-

• cot A/2 - tan A/2 = 2 cot A

Prove

Take L.H.S.

==> cot A/2 - tan A/2

[ ★ cot x = cos x /sin x

★ tan x = sin x /cos x ]

==> (cos A/2)/(sin A/2) - (sin A/2)/(cos A/2)

==> (cos ² A/2 - sin ² A/2 )/(cos A/2). (sin A/2)

[ ★ cos ² x - sin ² x = cos 2x

★ 2.sin x . cos x = sin 2x ]

So,

==> 2*(cos 2A/2)/(sin 2A/2)

==> 2 * cos A / sin A

[ ★ cos x/sin x = cot x ]

==> 2. cot A

= R.H.S.

That's proved.

_________________________

Some Important Formula

★ Cos 2x = cos ² x - sin ² x

= 1 - 2 sin ² x

= 2 cos ² x - 1

= ( 1 - tan ² x )/(1 + tan ² x)

★ Sin 2x = 2 Sin x Cos x

_________________________