Verify the following identities and then circle the letter o

Verify the following identities and then circle the letter of the correct answer 3 cot^2(t)/csc(t) a. 3 - 3 cos^2(t)/cos(t) b. 3 + 3 cos^2(t)/cos(t) c. 3 - 3 sin^2(t)/sin(t) d. -1 - sin^2(t)/3 sin(t) sin^2(x)/1 - cos(x) a. 1 + cos x b. cot x sin x + tan x

Solution

1. The answer is (b) 3-3sin^2t/ sin t

Given 3 cot^2 t /csc t

we know that cot t = cos t/sin t and

cosec t = 1 / sin t

substituting this in the given question

3(cos t/sin t)^2 / (1/sin t)

3 cos ^2 t * sin t / sin^2 t

3cos^2t * sin t / sin t * sin t

we know that

sin^2t+cos^2t=1

cos^2 t = 1 - sin ^ 2 t

on substituting this we get (b) 1 - sin ^2 t / sin t

2 . sin^2x/1-cosx

we know that sin^2 x = 1- cos^2x

cos x = 1/sec x

substituting this in the given question

1- cos^2x / cosx

1 - (1 / sec x)^2 /(1- 1/secx)

(1 - 1 / sec^2x) / (1- 1 /secx)

((sec^2x -1 )/sec^2 x) * sec x / (sec x -1)

sec ^ 2 x - 1 / (sec x- 1 ) sec x

(sec x + 1 )(sec x - 1) / (sec x-1) sec x

= sec x + 1 / sec x which is the required solution.