Question

Taking A = 30°, verify that
(i) cos4 A - sin4 A = cos 2A
(ii) 4 cos A cos (60° - A) cos (60° + A) = cos 3A.​

Answer 1

EXPLANATION.

Taking A = 30°.

(1) = Cos⁴A - Sin⁴A = Cos 2A.

Put the value of A = 30°.

→ Cos⁴(30°) - Sin⁴(30°) = Cos 2(30°).

→ ( √3/2)⁴ - ( 1/2)⁴ = Cos 60°.

→ 9/16 - 1/16 = Cos 60°.

→ 8/16 = Cos 60°.

→ 1/2 = 1/2. = Proved.

(2) = 4 Cos A Cos ( 60° - A) Cos ( 60° + A) =

Cos 3A.

→ put the value of A = 30° in equation.

→ 4 Cos (30°) Cos ( 60° - 30° ) Cos ( 60° + 30°)

= Cos 3(30°).

→ 4 ( √3/2) Cos ( 30°) Cos ( 90°) = Cos 90°.

→ 2√3 X √3/2 X 0 = 0.

→ 0 = 0 = Proved.

Answer 2

$\large\sf\underline{ \underline{ \red{solution : }}}$

$\huge \boxed{ \sf{(i)}}$

cos⁴A - sin⁴A = cos2A

cos⁴30 - sin⁴30 = cos2(30)

cos⁴30 - sin⁴30 = cos60

$\\ \sf{ {( \frac{\sqrt{3} }{2} )}^{4} - { (\frac{1}{2} )}^{4} =\frac{1}{2} } \\ \\ \\ \sf{ \frac{9}{16} - \frac{1}{16} = \frac{1}{2} } \\ \\ \\ \sf{ \frac{ \cancel{{8}} \: \: ^{1} }{ \cancel {16} \: \: ^{2} } } = \frac{1}{2} \\ \\ \\ \implies \sf{ \frac{1}{2} = \frac{1}{2} \: \: \: \: \: \: \: \: \: \: \pink{verified}}$

$\\ \\ \huge \boxed{ \sf{(ii)}}$

4cosA . cos(60 - A)cos(60+A) = cos3A

4cosA . cos(60-30)cos(60+30) = cos3(30)

4cosA . cos30 . cos90 = cos90

ㅤㅤㅤㅤㅤㅤㅤ[cos 90 = 0]

$\sf{0 = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \pink{verified}$