Question

if A=pi/8 than the value of (cos5Acos4A+sin5Asin4A)=​

Answer 1

$\small{ \green{ \bf \: cos5A. \: cos4A + sin5A. \: sin4A = cos \frac{\pi}{8} }}$

Step-by-step explanation:

We know that,

$\bf \: cos(x - y) = cos \: x .\: cos \: y + sin \: x. \: sin \: y$

Now given,

$\bf \: cos5A.cos4A + sin5A.sin4A \\ = \bf \: cos(5A - 4A) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ = \bf \: cosA \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ = \bf \: cos \frac{\pi}{8} \: \: \: \: \pink{\{ \because\bf \: A = \frac{\pi}{8} \}} \: \: \: \: \: \: \: \: \: \\ \bf$

Answer 2

Given :- A = π/8 .

To Find :- value of (cos5Acos4A + sin5Asin4A) = ?

Formula used :- cos(C - D) = cos C * cos D + sin C * sin D .

Solution :-

Let 5A = C and 4A = D .

so,

→ cos5A * cos4A + sin5A * sin4A

→ cos C * cos D + sin C * sin D

using above told formula we get,

→ cos (C - D)

putting C = 5A and D = 4A now,

→ cos (5A - 4A)

→ cos (A)

now, using cos 2A = (2 cos²A - 1)

→ cos 2A = 2 cos²A - 1

→ 2 cos²A = cos 2A + 1

→ cos²A = (cos 2A + 1)/2

→ cos A = √[(cos 2A + 1)/2]

finally putting given value of A = π/8 we get,

→ cos π/8 = √[(cos 2(π/8) + 1)/2]

→ cos π/8 = √[(cos π/4 + 1)/2]

putting cos π/4 = (1/√2)

→ cos π/8 = √[(1/√2 + 1)/2]

→ cos π/8 = √[(1 + √2)/2√2]

→ cos π/8 = √[√2(1 + √2)/2*√2*√2]

→ cos π/8 = √[(2 + √2)/4]

→ cos π/8 = (1/2)√(2 + √2) (Ans.)

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