Here are few Important Strategies that can be applied to simplify trigonometric expressions.
• Re-write expression in terms of sine and cosine
• Factor and then simplify
• Rationalize using conjugate of expression
• Simplify taking common denominator

Example 3: Re-write in terms of sine and cosine to simplify
(a) \(\frac{tan}{sec θ}\) (b) \(\frac{1 + tan θ}{1 + cot θ}\)
Solution 3: Re-write in terms of sine and cosine to simplify.
(a) \(\frac{tan θ}{sec θ}\) = \(\frac{sin θ/cos θ}{1/cos θ}\)
= \(\frac{sin θ}{cos θ}× \frac{cos θ}{1} \)
= sin
(b) \(\frac{1 + tan θ}{1 + cot θ}\) = \(\frac{1 + \frac{sin θ}{cos θ}}{1 + \frac{cos θ}{sin θ}}\)

= \(\frac{\frac{cos θ + sin θ}{cos θ}}{\frac{sin θ + cos θ}{sin θ}}\)
= \(\frac{sin θ}{cos θ}\) = tan

Example 4: Factor and then simplify
(a) \(\frac{cos^2θ + cos θ}{sin^2θ}\) (b) \(\frac{1 + sain^2x}{1 - sin^4x}\)
Solution 4: Factor and then simplify
(a )\(\frac{cos^2 + cos θ}{sin^2}\)
\(\frac{cos^2θ + cos θ}{sin^2 θ}\) = \(\frac{cos θ(cos θ + 1}{1-cos^2θ}\)
= \(\frac{cos θ(cos θ+ 1}{(1-cos θ)(1 - cos θ)}\) = \(\frac{cos θ}{1 - cos θ}\)
(b) \(\frac{1 + sain^2x}{1 - sin^4x}\)
= \(\frac{1 + sain^2x}{1 - sin^4x}\)
= \(\frac{1 + sain^2x}{(1-sin^2x)(1-sin^2x)}\)
= \(\frac{1}{1 - sin^2x}\) = \(\frac{1}{cos^2x}\) = sec^2x

Example 5: Rationalize using conjugate of an expression
(a) \(\sqrt{\frac{1+cos x}{1-cos x}}\) (b) \(\frac{1}{cos x - tan x}\)
Solution 5: Rationalize using conjugate of an expression
(a) \(\sqrt{\frac{1 + cos x}{1 - cos x}× \frac{1 + cos x}{1 + cos x}}\)
= \(\sqrt{\frac{(1 + cos x)^2}{1 - cos^2x}}\)
= \(\sqrt{\frac{(1 + cos x)^2}{sin^2x}}\)
=|\(\frac{1 + cos x}{sin x}\)|

(b) \(\frac{1}{cos x - tan x}\)
= \(\frac{1}{sec x - tan x} × \frac{sec x + tan x}{sec x + tan x}\)
= \(\frac{sec x + tan x}{sec^2x - tan^2 x}\)
= sec + tan
= \(\frac{1}{cos x} + \frac{sin x}{cos x}\)
= \(\frac{1 + sin x}{cos x}\)

Example 6: Simplify taking common denominator
(a) \(\frac{1}{1 - sin x} + \frac{1}{1 + sin x}\) (b) \(\frac{1 - sinθ }{cos θ } + \frac{cos θ}{1 - sin θ }\)
Example 6: Simplify taking common denominator

(a) \(\frac{1}{1 - sin x} + \frac{1}{1 + sin x}\)
= \(\frac{(1 + sin x) + (1 - sin x)}{(1 - sin x)(1 + sin x)}\) = \(\frac{2}{1 - sin^2x}\) = \(\frac{2}{cos^2x}\) = \({2 sec^2 }\)

(b) \(\frac{1 - sin θ}{cos θ} + \frac{cos θ}{1 - sin θ}\) = \(\frac{(1 - sin θ)^2 + cos^2θ}{cos θ(1- sin θ}\) = \(\frac{1 - 2sinθ + sin^2 + cos^2}{cos(1 - sin)}\)
= \(\frac{1-2sin + 1}{cosθ (1- sin θ)}\) = \(\frac{2- 2 sin}{cos(1- sin θ)}\) = \(\frac{2(1-sin θ)}{cos θ(1 - sin θ)}\) = \(\frac{2}{cos θ}\) = 2sec θ