Question

Show that cos square (45degree-theta)cos square (45degree+theta)=1

Answer 1

Question:

Show that cos²(45°-θ).cos²(45°+θ) = 1.

Formula used:

$\bold{Cos²\theta = \dfrac{1+Cos2\theta}{2}}$

Proof:

$\sf{LHS = cos²(45°-θ).cos²(45°+θ)}$

$\sf{LHS = \bigg(\dfrac{1+cos2(45°-\theta )}{2}\bigg)\bigg(\dfrac{1+cos2(45°+\theta )}{2}\bigg)}$

$\sf{LHS = \bigg(\dfrac{1+cos(90°-2\theta )}{2}\bigg)\bigg(\dfrac{1+cos(90°+2\theta )}{2}\bigg)}$

$\sf{LHS = \bigg(\dfrac{1+sin2\theta }{2}\bigg)\bigg(\dfrac{1-sin2\theta }{2}\bigg) }$

$\sf{LHS = \dfrac{1-sin²2\theta }{4}}$$\sf{=\dfrac{cos²2\theta }{4}≠RHS}$