Mathematics

Mathematics

Problem

Simplify the expression 1+tan⁡2θsec⁡2θ\dfrac{1+\tan^2\theta}{\sec^2\theta}. a) tan⁡2θ\tan^2\theta b) cos⁡2θ\cos^2\theta c) 11 d) cot⁡2θ\cot^2\theta
Intuitive explanation

The key here is recognizing a fundamental trigonometric identity: 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta. Once you spot that the numerator is exactly this identity, the fraction collapses immediately. This is a great reminder that knowing your core identities lets you simplify what looks complicated into something trivial.

Intuitive explanation

The key here is recognizing a fundamental trigonometric identity: 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta. Once you spot that the numerator is exactly this identity, the fraction collapses immediately. This is a great reminder that knowing your core identities lets you simplify what looks complicated into something trivial.

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