I need help in understanding how to identify which identity to use. using this question as an example.
Show that tan theta (sin theta+cot theta cos theata)all over cos theta = sec^2 theta. Please explain.
tan(sin+cot*cos)/cos
When dealing with such a combination of functions, it is usually good to start out by converting everything to sin and cos:
(sin/cos)(sin+(cos/sin)cos)/cos
sin/cos * (sin^2 + cos^2)/sin * 1/cos
(sin^2 + cos^2)/cos^2
1/cos^2
sec^2
To show that the given expression is equal to sec^2(theta), we can start by simplifying the left side of the equation using trigonometric identities.
First, let's expand the numerator:
tan(theta) (sin(theta) + cot(theta) cos(theta))
Now, let's simplify each term:
tan(theta) = sin(theta) / cos(theta) (Using the definition of tan(theta))
cot(theta) = cos(theta) / sin(theta) (Reciprocal property: cot(theta) = 1 / tan(theta))
Replacing these values in the expression:
(sin(theta) / cos(theta)) * (sin(theta) + (cos(theta) / sin(theta))) * cos(theta)
To simplify further, let's find a common denominator for the two fractions:
(sin^2(theta) + cos^2(theta)) / cos(theta) (Expanding the numerator)
1 / cos(theta) (Using the trigonometric identity: sin^2(theta) + cos^2(theta) = 1)
Finally, simplifying further:
sec^2(theta) (Using the definition of sec^2(theta))
Therefore, we have shown that tan(theta) (sin(theta) + cot(theta) cos(theta)) / cos(theta) = sec^2(theta).
In summary, to solve this problem, we used the definitions and properties of trigonometric functions, as well as trigonometric identities to simplify the given expression and arrive at the desired result.