Video narration
Given sec A plus tan A equals x, it feels impossible: one equation, two unknowns. The fix is adding and isolating-but where does the second equation come from? It is hiding in sec2A minus tan2A equals 1. Factor it: (sec A plus tan A)(sec A minus tan A) equals 1. Now the problem flips. If sec A plus tan A equals x, its reciprocal partner is sec A minus tan A equals 1/x, with x not zero. Add the equations: 2 sec A equals x plus 1/x, so sec A equals (x2 plus 1) over 2x. Subtract them: 2 tan A equals x minus 1/x, giving tan A equals (x2 minus 1) over 2x. For x equals 3, sec A is 5 over 3 and tan A is 4 over 3. Remember: create the partner, add for sec A, subtract for tan A.