Prove the trigonometric identity: ( heta)/(1 - heta) = extsec heta + exttan heta

Question

ScanSolve AI Tutor · Accepted Answer

Here's the solution for question 9(a).

Start with the left-hand side (LHS) of the identity.
$\text{LHS} = \frac{\cos \theta}{1 - \sin \theta}$
Multiply the numerator and denominator by the conjugate of the denominator, which is $1 + \sin \theta$ .
$\text{LHS} = \frac{\cos \theta}{1 - \sin \theta} \times \frac{1 + \sin \theta}{1 + \sin \theta}$
Expand the numerator and simplify the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .
$\text{LHS} = \frac{\cos \theta (1 + \sin \theta)}{1^2 - \sin^2 \theta}$ $\text{LHS} = \frac{\cos \theta + \cos \theta \sin \theta}{1 - \sin^2 \theta}$
Apply the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ , which means $1 - \sin^2 \theta = \cos^2 \theta$ .
$\text{LHS} = \frac{\cos \theta + \cos \theta \sin \theta}{\cos^2 \theta}$
Separate the fraction into two terms and simplify.
$\text{LHS} = \frac{\cos \theta}{\cos^2 \theta} + \frac{\cos \theta \sin \theta}{\cos^2 \theta}$ $\text{LHS} = \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta}$
Express the terms using the definitions of $\sec \theta$ and $\tan \theta$ .
$\text{LHS} = \sec \theta + \tan \theta$
This matches the right-hand side (RHS) of the identity.

The identity is proven.

Prove the trigonometric identity: ( heta)/(1 − heta) = extsec heta + exttan heta