Trigonometric identities - 6

Trigonometric identity:  1 + [ tan²A / (1 + sec A) ] = sec A

We can see that all the trigonometric functions in this identity can be simplified to sine and cosine functions. The easiest way to solve such identities is to do exactly that. On the other hand, tan²A can be simplified to (sec²A - 1) (this is derived from the identity 1 + tan²A = sec²A). Going with the latter approach, the LHS can be written as

1 + [ (sec²A - 1) / (1 + sec A) ]

(sec²A - 1) can be factored, as it is a difference of two squares, as follows:

sec²A - 1 = sec²A - 1² = (sec A + 1)(sec A - 1)

Replace (sec²A - 1) with (sec A + 1)(sec A - 1) in the LHS,

1 + [ (sec A + 1)(sec A - 1) / (1 + sec A) ]

(sec A + 1) is common in both the numerator and denominator, so cancel it out,

1 + [ (sec A - 1) ]

= 1 + sec A - 1

= sec A

... which is the RHS expression

Trigonometric identities applied above

• 1 + tan²A = sec²A