Is cos((5θ)/2)cos(θ/2)= 1/4 cos^2(5θ) ?

The answer to whether cos((5θ)/2)cos(θ/2)= 1/4 cos^2(5θ) is False

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Popular >

prove cos((5θ)/2)cos(θ/2)= 1/4 cos^2(5θ)

Solution

prove

cos (\frac{5 θ}{2}) cos (\frac{θ}{2}) = \frac{1}{4} cos^{2} (5 θ)

Solution

False

Solution steps

cos (\frac{5 θ}{2}) cos (\frac{θ}{2}) = \frac{1}{4} cos^{2} (5 θ)

Show that the two sides are not equal

For

cos (\frac{5 θ}{2}) cos (\frac{θ}{2}) = \frac{1}{4} cos^{2} (5 θ)

plug in

θ = 0

cos (\frac{5 \cdot 0}{2}) cos (\frac{0}{2}) = 1

cos (\frac{5 \cdot 0}{2}) cos (\frac{0}{2})

Simplify:

\frac{5 \cdot 0}{2} = 0

\frac{5 \cdot 0}{2}

Apply rule

0 \cdot a = 0

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

Simplify:

\frac{0}{2} = 0

\frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

cos (0) cos (0) = cos^{2} (0)

cos (0) cos (0)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (0) cos (0) = cos^{1 + 1} (0)

= cos^{1 + 1} (0)

Add the numbers:

1 + 1 = 2

= cos^{2} (0)

= cos^{2} (0)

Use the following trivial identity:

cos (0) = 1

cos (0)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

= 1^{2}

Simplify

= 1

\frac{1}{4} cos^{2} (5 \cdot 0) = \frac{1}{4} (Decimal : 0.25)

\frac{1}{4} cos^{2} (5 \cdot 0)

Simplify:

5 \cdot 0 = 0

5 \cdot 0

Apply rule

0 \cdot a = 0

= 0

\frac{1}{4} cos^{2} (0) = \frac{cos ^{2} ( 0 )}{4}

\frac{1}{4} cos^{2} (0)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ^{2} ( 0 )}{4}

Multiply:

1 \cdot cos^{2} (0) = cos^{2} (0)

= \frac{cos ^{2} ( 0 )}{4}

= \frac{cos ^{2} ( 0 )}{4}

Use the following trivial identity:

cos (0) = 1

cos (0)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

= \frac{1 ^{2}}{4}

Simplify

= \frac{1}{4}

The two sides are not equal

\Rightarrow False

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Frequently Asked Questions (FAQ)

Is cos((5θ)/2)cos(θ/2)= 1/4 cos^2(5θ) ?
The answer to whether cos((5θ)/2)cos(θ/2)= 1/4 cos^2(5θ) is False