Question

show that cos 2 (45 + theta) + cos 2 (45 - theta) / tan (60 + theta) tan (30 - theta) = 1

Answer 1

Solution:

To Show:

$\dfrac{cos^2(45+\theta)+cos^2(45-\theta)}{tan(60+\theta).tan(30-\theta)}=1$

Formulas Used:

$cosA=sin(90-A)\\\\tanA=cot(90-A)\\\\cos^2A+sin^2A=1\\\\tanA.cotA=1$

Proof:

Taking Left Hand Side,$\dfrac{cos^2(45+\theta)+cos^2(45-\theta)}{tan(60+\theta).tan(30-\theta)}\\\\\\=\dfrac{cos^(45+\theta)+sin^2(90-45+\theta)}{tan(60+\theta).cot(90-30+\theta)}\\\\\\=\dfrac{cos^2(45+\theta)+sin^2(45+\theta)}{tan(60+\theta).cot(60+\theta)}\\\\\\=\dfrac{1}{1}\\\\\\=1$

Right Hand Side = Left Hand Side.

Hence Proved.

Answer 2

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