AM-2698

Basic trigonometric identities

A Math · Secondary 4 · Hard · Basic trigonometric identities

1. Use $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ so $1+\tan^2\theta=1+\dfrac{\sin^2\theta}{\cos^2\theta}=\dfrac{\sin^2\theta+\cos^2\theta}{\cos^2\theta}=\dfrac{1}{\cos^2\theta}=\sec^2\theta$.
2. Use $1+\tan^2\theta=\dfrac{1+\sin^2\theta}{\cos^2\theta}=\dfrac{2}{\cos^2\theta}$, so it equals $2\sec^2\theta$.
3. Use $\tan^2\theta=\sec^2\theta-2$, so $1+\tan^2\theta=\sec^2\theta-1$.
4. Since $\tan\theta=\sec\theta$, clearly $1+\tan^2\theta=1+\sec^2\theta$.

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