Fundamental Trigonometric Identities

Fundamental Trigonometric Identities

Fundamental identities allow you to rewrite trig expressions without a calculator. Key relationships include $\sin^2\theta+\cos^2\theta=1$, $\tan\theta=\sin\theta/\cos\theta$, and reciprocal ratios. In a proof, change only one side at a time and write an identity beside each transformation; the aim is to show equivalence, not to make both sides look complicated.

Worked reasoning

Simplify $\frac{1-\sin^2\theta}{\cos\theta}$. Since $1-\sin^2\theta=\cos^2\theta$, the expression becomes $\cos^2\theta/\cos\theta=\cos\theta$.

Exam method

1. Choose one side to work on. 2. Replace a recognisable identity. 3. Factor/cancel only common factors and state the final equivalent form.

Proof writing earns the method marks

In a trig proof, begin with the side that looks more complicated and label each identity used in a margin or after an equals sign. Avoid changing both sides at once: that proves neither side reaches the other. Factor before cancelling, and state any denominator restriction if relevant. At the finish, make the target expression appear exactly as printed. Equivalent but unsimplified forms can hide whether the intended identity has actually been shown.

One-minute retrieval: Fundamental Trigonometric Identities

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.