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Math  /  Trigonometry

Question2. Prove: 1+tan⁡2θ=1cos⁡2θ1+\tan ^{2} \theta=\frac{1}{\cos ^{2} \theta} (4 marks)

Studdy Solution

STEP 1

1. We are working with trigonometric identities.
2. We need to use known trigonometric identities to prove the given equation.

STEP 2

1. Recall and use the Pythagorean identity for tangent and secant.
2. Manipulate the identity to match the given equation.

STEP 3

Recall the Pythagorean identity for tangent and secant:
1+tan⁡2θ=sec⁡2θ 1 + \tan^2 \theta = \sec^2 \theta
This identity is derived from the fundamental Pythagorean identity sin⁡2θ+cos⁡2θ=1 \sin^2 \theta + \cos^2 \theta = 1 .

STEP 4

Express the secant function in terms of cosine:
sec⁡θ=1cos⁡θ \sec \theta = \frac{1}{\cos \theta}
Therefore,
sec⁡2θ=(1cos⁡θ)2=1cos⁡2θ \sec^2 \theta = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta}

STEP 5

Substitute the expression for sec⁡2θ\sec^2 \theta back into the Pythagorean identity:
1+tan⁡2θ=sec⁡2θ 1 + \tan^2 \theta = \sec^2 \theta 1+tan⁡2θ=1cos⁡2θ 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta}
This matches the given equation, thus proving the identity.
The identity 1+tan⁡2θ=1cos⁡2θ 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} is proven.

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