Question

Prove that (1+cos pi/4)(1+cos 3pi/4)(1+cos 5pi/4) (1+cos 7pi/4) =1/4
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Answer 1

Step-by-step explanation:

refer to the attachment

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Answer 2

Given:

A trigonometric equation (1+cos π/4)(1+cos 3π/4)(1+cos 5π/4) (1+cos 7π/4) =1/4.

To Find:

The proof of the given equation.

Solution:

The given problem can be solved using the concepts of trigonometry.

1. The given equation is (1+cos π/4)(1+cos 3π/4)(1+cos 5π/4) (1+cos 7π/4) =1/4.

2. Consider the LHS of the given equation,

=>(1+cos π/4)(1+cos 3π/4)(1+cos 5π/4) (1+cos 7π/4),

=> Cos(π/4) = 1/√2,

=> Cos(3π/4) = Cos(π-π/4) = -1/√2,

=> Cos(5π/4) = Cos(π+π/4) = -1/√2,

=> Cos(7π/4) = Cos(2π-π/4) = 1/√2,

3. Substitute the values of the angles mentioned above,

=> ( 1 + 1/√2) x ( 1 - 1/√2 ) x ( 1 - 1/√2) x ( 1 + 1/√2),

=> (1-1/2) x ( 1-1/2), [(a+b) x (a-b) = a²-b²]

=> 1/2 x 1/2,

=> 1/4 = LHS.

Hence proved.

Therefore, the equation (1+cos π/4)(1+cos 3π/4)(1+cos 5π/4) (1+cos 7π/4) =1/4 is proved.