Question

Show that cos?(45°+theta)+cos^2(45° - theta )/tan(60° + theta) tan(30° - theta)=1​

Answer 1

Answer:

os

2

(45+θ)+cos

2

(45−θ)

=1

Formulas Used:

\begin{gathered}cosA=sin(90-A)\\\\tanA=cot(90-A)\\\\cos^2A+sin^2A=1\\\\tanA.cotA=1\end{gathered}

cosA=sin(90−A)

tanA=cot(90−A)

cos

2

A+sin

2

A=1

tanA.cotA=1

Proof:

Taking Left Hand Side,\begin{gathered}\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\end{gathered}

tan(60+θ).tan(30−θ)

cos

2

(45+θ)+cos

2

(45−θ)

=

tan(60+θ).cot(90−30+θ)

cos

(

45+θ)+sin

2

(90−45+θ)

=

tan(60+θ).cot(60+θ)

cos

2

(45+θ)+sin

2

(45+θ)

=

1

1

=1

Right Hand Side = Left Hand Side.

Hence Proved.