Verify trig identities using simplification techniques

Question

1. Csc x - sin x = cos x cot x
2. Tan² o / 1+ sec o = 1- cos o / cos o
3. csc a + cot a = sin a / 1- cos a

Answer 1

csc x - sin x = cos x cot x Will manipulate the left side - Turn csc into sin

1/sin x - sin x Get a common denominator

1/sinx - sin x/sin x

(1-sin²x)/sin x Pythagorean identities, 1-sin²x sub in cos²x

cos²x/sin x Split up to cos²x

(cos x)(cos x/sin x) cos x/sin x = cot x

cos x cot x = cos x cot x

tan²x/(1+sec x) = (1-cos x)/cos x Will manipulate the left side using x instead of theta for the variable for ease of typing, Pythagorean identities tan²x=sec²x-1

(sec²x-1)/(1+sec x) Factor the numerator, it is difference of squares

(sec x+1)(sec x -1)/(1+sec x) (sec x+1) and (1+sec x) cancel out

(sec x -1) Change sec to cos

1/cos x - 1 Get a common denominator

1/cos x - cos x /cos x

(1-cos x)/cos x = (1-cos x)/cos x