It's a well-known identity that tan²θ+1 ≡ sec²θ, so the equation is true, regardless of the value of θ.

I like how you've branched the puzzles into different categories though. ^_^

Ah. OK then.

All I know is that sin 30θ = 1/2 and that cosθ is √3/2, but I can work everything out from that.

tanθ = sinθ ÷ cosθ = (1/2)/(√3/2) = 1/√3
Therefore, tan²θ = 1/3.

secθ = 1 ÷ cosθ = 1/(√3/2) = 2/√3
Therefore, sec²θ = 4/3.

Subsituting these into the given equation gives 1/3 + 1 = 4/3, which is obviously true.

T # 2

What is the value of (1-CosA) / (1+CosA) given that tan A = 3/4?

krassi_holmz, what a guess!
Anyhow, this is how it's solved.
tanA = Opposite side/Adjacent side=3/4,
Therefore, Hypotenuse=[(3²+4²)]⊃½=5,
therefore, SinA=3/5, CosA=4/5,
(1-CosA)/((1+CosA) = (1/5)/(9/5) = 1/9.

That was a great guess, krassi_holmz!

T # 3

From the top and bottom of a tower, the angle of elevation of the top of a cliff of height 400m are observed to be 30 degrees and 60 degrees.Determine the height of the tower.

Excellent! Good work, mathsyperson!

Factorising both sides gives:

cos²A(1-cos²A) = sin²A(1-sin²A)

However, using the identity that sin²θ + cos²θ = 1, we can replace (1-cos²A) with sin²A and (1-sin²A) with cos²A.

Therefore, both sides equal cos²Asin²A.

T # 5

Find the value of Sin 15°
(Given Sin45°=1/√2, Cos45°=1/√2, Sin30°=1/2, Cos30°=√3/2)

T # 7

Solve:-

Cos²θ/(Cot²θ-Cos²θ) = 3.

i would rather use a calculator for this prob.

ganesh wrote:

T # 7

Solve:-

Cos²θ/(Cot²θ-Cos²θ) = 3.

Not too sure how to do this but i manipulated the question until i got down to 4cos sin = 3 and then sin ≠ 3 and cos =3/4 I just dont know how to find this without a calculator.