Answer to Question #129038 in Trigonometry for venkat

Question

1. Prove that ( 1 - cos2 x ) cosec2 x = 1 .

2. Prove that (tan x sin x ) + cos x = sec x .

3. Prove that Tan^4 x + Tan^2 x = sec^4 x - sec^2 x .

4. prove that Cot x + Tan x = sec x cosec x.

5. Prove that (1 + sin x) (1 - sin x) = (sec x - Tan x)2 .

Answer 1

solution for question 2

(sinx/cosx .sinx) +cosx=

sin² x/cosx +cosx=

(sin²x +cos²x)/cosx=

1/cosx

= secx

solution for question 1

(1 -cos²x)cosec²x=

(sin²x)cosec²x=

(sin²x) . 1/sin²x=

= 1

solution for question 4

cotx +tanx =

cosx/sinx + sinx/cosx=

(cos²x +sin²x)/(cosxsinx)=

1/(cosxsinx)=

1/cosx .1/sinx=

=secxcosecx

question 5 solution

(1 +sinx)(1 -sinx)

1 -sin²x =cos²x from the difference of two squares

(1/cosx +sinx/cosx)(1/cosx -sinx/cosx)=

(secx+Tanx)(secx -Tanx)

looking at the reverse method

(secx -Tanx)² =(1 +sinx)(1 -sinx)

(1/cosx -sinx/cosx)

(1 -sinx)²/cos²x

question 3 solution

Tan⁴x + Tan²x = sec⁴x -sec²x

Tan²x(Tan²x +1)

recall cos²x + sin²x =1

1 +Tan²x =sec²x

(sec²x -1)(sec²x)

= sec⁴x -sec²x