Question

Prove the following trigonometric identity:
$\frac{cos θ}{1 + sin θ} + \frac{1 + sin θ}{cos θ} = 2 sec θ$

Approach

Prove $\frac{c o s θ}{1 + s i n θ} + \frac{1 + s i n θ}{c o s θ} = 2 sec θ$ . Combine over a common denominator: $\frac{c o s ^{2} θ + ( 1 + s i n θ ) ^{2}}{( 1 + s i n θ ) c o s θ}$ . Expand the numerator: $cos^{2} θ + 1 + 2 sin θ + sin^{2} θ = 2 + 2 sin θ = 2 (1 + sin θ)$ . Cancel to get $\frac{2}{c o s θ} = 2 sec θ$ .

Step-by-step working

1
Combine fractions over common denominator: $\frac{c o s ^{2} θ + ( 1 + s i n θ ) ^{2}}{( 1 + s i n θ ) c o s θ}$ .
1 mark
2
Expand numerator: $cos^{2} θ + 1 + 2 sin θ + sin^{2} θ = (cos^{2} θ + sin^{2} θ) + 1 + 2 sin θ = 2 + 2 sin θ$ .
0.5 marks
3
Factor: $2 (1 + sin θ)$ .
1 mark
4
Cancel: $\frac{2 ( 1 + s i n θ )}{( 1 + s i n θ ) c o s θ} = \frac{2}{c o s θ} = 2 sec θ$ .
0.5 marks

Concepts used

trigonometrytrigonometric identitiesprovesimplifysec theta

Approach

Step-by-step working

Concepts used

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