Simplify the trigonometric expression sec x tan2 xsec x 91

Simplify the trigonometric expression. sec x - tan^2 x/sec x 91 - cos x)^2 + sin^2 x/1 - cos x Verify the trigonometric identity. Csc(-x)-1/sin(-x)-1 = csc x

Solution

1)

a)secx -(tan²x_/secx)

=(secxsecx -tan²x)_/secx

=(sec²x -tan²x)_/secx

=1_/secx..................[since sec²x -tan²x =1]

=cosx

b)[(1-cosx)²+sin²x]_/(1-cosx)

=[(1²+cos²x-2cosx)+sin²x]_/(1-cosx)

=[1+cos²x-2cosx+sin²x]_/(1-cosx)

=[1+sin²x+cos²x-2cosx]_/(1-cosx)

=[1+1-2cosx]_/(1-cosx)........................since sin²x+cos²x =1

=[2-2cosx]_/(1-cosx)

=2(1-cosx)_/(1-cosx)

=2

2)[csc(-x)-1]_{/[sin(-x) -1]}

=[-csc(x)-1]_{/[-sin(x) -1]}

=[csc(x)+1]_{/[sin(x) +1]}

=[(1/sin(x))+1]_{/[sin(x) +1]}

=[(1+sin(x))/sinx]_{/[sin(x) +1]}

=1/sinx

=cscx