Question

Prove that:
Prove that : cos(45° + A) cos(45º - A) =
cOS 2A
cos 2A​

Answer 1

$\: \:$$\large \underline \bold{Solution :-}$

=> $\small \bold{Cos(45 + A) Cos(45 - A)}$

=> $\small \bold{Cos(90 - 45 + A) Cos(45 - A)}$

=> $\small \bold{Cos[90 - (45 - A)] Cos(45 - A)}$

=> $\small \bold{Sin(45 - A) Cos(45 - A)}$

=> $\small \bold{\dfrac{2Sin(45 - A)Cos(45 - A)}{2}}$

=> $\small \bold{\dfrac{Sin[2(45 - A)]}{2}}$

=> $\small \bold{\dfrac{Sin(90 - 2A)}{2}}$

=> $\small \bold{\dfrac{Cos 2A}{2}}$

$\: \: \:$$\large \bold{HENCE \: PROOF}$

$\small \underline \bold{Using \: Identity :-}$

$\: \: \:$$\small \bold{1) \: Cos(90 - x) = Sinx}$

$\: \: \:$$\small \bold{2) \: Sin(90 - x) = Cosx}$

$\: \: \:$$\small \bold{3) \: Sin 2x = 2SinxCosx}$

Answer 2

Answer:

We Should Know Some Trignometric Identities For Solving This Question.

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$\begin{gathered}1. \: \: \sin(2 \alpha ) = 2 \sin( \alpha ) \cos( \alpha ) \\\end{gathered} </p><p>1.sin(2α)=2sin(α)cos(α)$

$2. \: \: \sin(180 - \alpha ) = \sin( \alpha )2.sin(180−α)=sin(α)$

$3. \: \: \sin(90 - \alpha ) = \cos( \alpha )3.sin(90−α)=cos(α)$

━━━━━━━━━━━━━━━━━━━━━━━━━━

$L.H.S :- Sin10. Sin30. Sin50. Sin70$

$\begin{gathered}= \cos(90 - 10) \sin(30) \cos(90 - 40) \cos(90 - 70) \\ = \cos(80) \: \frac{1}{2} \: \cos(40) \: \cos(20) \\ = \frac{1}{4 \sin(20) } \cos(8 0 ) \cos(40) . \: 2 \sin(20) \cos(20) \\ = \frac{1}{8 \sin(20) } \cos(80) 2. \cos(40) \sin(40) \\ = \frac{1}{16 \sin(20) } 2 \cos(80) \sin(80 ) \\ = \frac{1}{16 \sin(20) } \sin(160) \\ = \frac{1}{16 \sin(20) } \sin(180 - 20) \\ = \frac{1}{16 \sin(20) } \sin(20) \\ = \frac{1}{16} \: \: \: \: \: \: \: \: .........R.H.S\end{gathered}$

$=cos(90−10)sin(30)cos(90−40)cos(90−70)$

$=cos(80) 2/1$

$cos(40)cos(20)= 4sin(20)1$

$cos(80)cos(40).2sin(20)cos(20)= 8sin(20)1$

$cos(80)2.cos(40)sin(40)= 16sin(20)1$

$2cos(80)sin(80)= 16sin(20)1$

$sin(160)= 16sin(20)1$

$sin(180−20)= 16sin(20)1$

$sin(20)= 16/1$

.........R.H.S,