1. (Tan ө + cos ө)/ (sec ө + cot ө)
2. (Tan ө + cot ө)/ sec ө
2. Try making that
(sin theta/cos theta)*cos theta
+ (cos theta/sin theta)*cos theta
= sin theta + (1 - sin^2 theta)/sin theta
= sin theta + cos theta cot theta
1. Try similar tricks with that one yourself. Write tan, cot and sec in terms of sin and cos
To simplify each expression, we can start by rewriting the trigonometric functions in terms of sine and cosine.
1. (Tan ө + cos ө)/ (sec ө + cot ө):
First, let's convert tangent and cotangent to their equivalent expressions using sine and cosine:
Tan ө = sin ө / cos ө
Cot ө = cos ө / sin ө
Now, substitute these into the original expression:
(sin ө / cos ө + cos ө) / (1 / cos ө + cos ө / sin ө)
Next, simplify the fractions by finding a common denominator:
[(sin ө + cos ө * cos ө) / cos ө] / [(sin ө + cos ө) / sin ө]
Now, we can simplify further by multiplying the numerator and denominator by the reciprocal of the second fraction:
[(sin ө + cos ө * cos ө) / cos ө] * [sin ө / (sin ө + cos ө)]
Expanding the expression, we get:
[(sin ө + cos ө * cos ө) * sin ө] / [(sin ө + cos ө) * cos ө]
Finally, we can simplify the numerator by using the trigonometric identity sin^2 ө + cos^2 ө = 1:
[sin ө + cos^2 ө * sin ө] / [(sin ө + cos ө) * cos ө]
This is the simplified form of the expression (Tan ө + cos ө)/ (sec ө + cot ө).
2. (Tan ө + cot ө) / sec ө:
Following a similar process as above, let's convert tangent and cotangent to their equivalent expressions using sine and cosine:
Tan ө = sin ө / cos ө
Cot ө = cos ө / sin ө
Now, substitute these into the original expression:
(sin ө / cos ө + cos ө / sin ө) / (1 / cos ө)
Next, simplify the numerator by finding a common denominator:
[(sin ө + cos ө * cos ө) / cos ө] / (1 / cos ө)
Now, we can simplify further by multiplying the numerator and denominator by the reciprocal of the second fraction:
[(sin ө + cos ө * cos ө) / cos ө] * [cos ө / 1]
Expanding the expression, we get:
[(sin ө + cos ө * cos ө) * cos ө] / cos ө
Finally, we can simplify the numerator by using the trigonometric identity sin^2 ө + cos^2 ө = 1:
[sin ө * cos ө + cos^3 ө] / cos ө
This is the simplified form of the expression (Tan ө + cot ө) / sec ө.