16. write the expression tan x sin x in terms of cos x.\n17. write the expression \\( \\frac { \\tan ^ { 2 }…

16. write the expression tan x sin x in terms of cos x.\n17. write the expression \\( \\frac { \\tan ^ { 2 } x } { 1 + \\tan ^ { 2 } x } \\) in terms of sin x.\n18. simplify \\( \\frac { \\cos x } { 1 - \\sin x } - \\tan x \\).\n19. simplify \\( \\frac { 1 } { \\sin ^ { 2 } x } - \\frac { 1 } { \\tan ^ { 2 } x } \\).\n20. simplify \\( \\frac { \\cos x } { \\tan x } + \\sin x \\).\n21. simplify \\( 1 + \\frac { 1 } { \\tan ^ { 2 } x } \\).\n22. simplify \\( \\frac { 1 - \\frac { 1 } { \\cos ^ { 2 } x } } { 1 - \\cos ^ { 2 } x } \\).\n23. prove that \\( \\tan x + \\frac { 1 } { \\tan x } = \\frac { 1 } { \\sin x \\cos x } \\).

16. write the expression tan x sin x in terms of cos x.\n17. write the expression \\( \\frac { \\tan ^ { 2 } x } { 1 + \\tan ^ { 2 } x } \\) in terms of sin x.\n18. simplify \\( \\frac { \\cos x } { 1 - \\sin x } - \\tan x \\).\n19. simplify \\( \\frac { 1 } { \\sin ^ { 2 } x } - \\frac { 1 } { \\tan ^ { 2 } x } \\).\n20. simplify \\( \\frac { \\cos x } { \\tan x } + \\sin x \\).\n21. simplify \\( 1 + \\frac { 1 } { \\tan ^ { 2 } x } \\).\n22. simplify \\( \\frac { 1 - \\frac { 1 } { \\cos ^ { 2 } x } } { 1 - \\cos ^ { 2 } x } \\).\n23. prove that \\( \\tan x + \\frac { 1 } { \\tan x } = \\frac { 1 } { \\sin x \\cos x } \\).

Answer

Explanation:

Step1: Use the identity (\tan x=\frac{\sin x}{\cos x})

Substitute (\tan x=\frac{\sin x}{\cos x}) into (\tan x\sin x). We get (\frac{\sin x}{\cos x}\cdot\sin x=\frac{\sin^{2}x}{\cos x}).

Step2: Use the Pythagorean identity (\sin^{2}x = 1-\cos^{2}x)

Substitute (\sin^{2}x = 1-\cos^{2}x) into (\frac{\sin^{2}x}{\cos x}). We have (\frac{1 - \cos^{2}x}{\cos x}=\frac{1}{\cos x}-\cos x).

Answer:

(\frac{1}{\cos x}-\cos x)