Prove that each of the following identities is true:?

= { 1 / [1 + cos(x)] } + { [1 / [1 - cos(x)] }

= { [1 - cos(x)] + [1 + cos(x)] } / { [1 + cos(x)].[1 - cos(x)] }

= [1 - cos(x) + 1 + cos(x)] / [1 - cos²(x)]

= 2 / [1 - cos²(x)] → you know that: cos²(x) + sin²(x) = 1 → sin²(x) = 1 - cos²(x)

= 2 / sin²(x)

= 2 * [1/sin²(x)]

= 2 * [1/sin(x)]² → you know that: 1/sin(x) = csc(x)

= 2.csc²(x)


= [1 - sec(x)] / [1 + sec(x)] → you know that: sec(x) = 1/cos(x)

= [1 - {1/cos(x)}] / [1 + {1/cos(x)}]

= [{cos(x)/cos(x)} - {1/cos(x)}] / [{cos(x)/cos(x)} + {1/cos(x)}]

= [{cos(x) - 1}/cos(x)] / [{cos(x) + 1}/cos(x)] → you can simplify by cos(x)

= [{cos(x) - 1}] / [{cos(x) + 1}]

= [cos(x) - 1] / [cos(x) + 1]


= cos(t) / [1 + sin(t)] → you multiply by cos(t) the top and the bottom

= [cos(t) * cos(t)] / {[1 + sin(t)] * cos(t)}

= cos²(t) / {[1 + sin(t)].cos(t)} → you know that: cos²(t) + sin²(t) = 1 → cos²(t) = 1 - sin²(t)

= [1 - sin²(t)] / {[1 + sin(t)].cos(t)} → you recognize: a² - b² = (a + b).(a - b)

= {[1 + sin(t)].[1 - sin(t)} / {[1 + sin(t)].cos(t)} → you can simplify by [1 + sin(t)]

= {[1 - sin(t)]} / {cos(t)}

= [1 - sin(t)] / cos(t)


= [1 - sin(t)]² / cos²(t) → you know that: cos²(t) + sin²(t) = 1 → cos²(t) = 1 - sin²(t)

= [1 - sin(t)]² / [1 - sin²(t)] → you recognize: a² - b² = (a + b).(a - b)

= [1 - sin(t)]² / { [1 - sin(t)].[1 + sin(t)] }

= { [1 - sin(t)].[1 - sin(t)] } / { [1 - sin(t)].[1 + sin(t)] } → you can simplify by [1 - sin(t)]

= [1 - sin(t)] / [1 + sin(t)] → you made a mistake


= [sec(x) + 1] / tan(x) → you know that: sec(x) = 1/cos(x)

= [{1/cos(x)} + 1] / tan(x) → you know that: tan(x) = sin(x)/cos(x)

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

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

= [ {1 + cos(x)}/cos(x) ] / [sin(x)/cos(x)] → you can simplify by cos(x)

= [1 + cos(x)] / sin(x) → you divide by [1 + cos(x)] the top and the bottom

= { [1 + cos(x)] / [1 + cos(x)] } / { sin(x) / [1 + cos(x)] } → you simplify

= 1 / { sin(x) / [1 + cos(x)] } → you multiply by tan(x) the top and the bottom

= tan(x) / { tan(x).sin(x) / [1 + cos(x)] } → you know that: tan(x) = sin(x)/cos(x)

= tan(x) / { {sin(x)/cos(x)}.sin(x) / [1 + cos(x)] }

= tan(x) / { {sin²(x)/cos(x)} / [1 + cos(x)] }

= tan(x) / { sin²(x) * [1/cos(x)] / [1 + cos(x)] } → you know that: sin²(x) = 1 - cos²(x)

= tan(x) / { [1 - cos²(x)] * [1/cos(x)] / [1 + cos(x)] } → you know that: 1 - cos²(x) = [1 + cos(x)].[1 - cos(x)]

= tan(x) / { [1 + cos(x)].[1 - cos(x)] * [1/cos(x)] / [1 + cos(x)] } → you can simplify by [1 + cos(x)]

= tan(x) / { [1 - cos(x)] * [1/cos(x)] }

= tan(x) / { [1/cos(x)] - [cos(x)/cos(x)] }

= tan(x) / { [1/cos(x)] - 1 } → you know that: 1/cos(x) = sec(x)

= tan(x) / { [sec(x)] - 1 }

= tan(x) / [sec(x) - 1]

1.
L.H.S.
= [1 / (1 + cos x)] + [1 / (1 - cos x)]
= [(1 - cos x) / (1 + cos x)(1 - cos x)] + [(1 + cos x) / (1 + cos x)(1 - cos x)]
= [(1 - cos x) + (1 + cos x)] / [(1 + cos x)(1 - cos x)]
= [1 - cos x + 1 + cos x] / [1 - cos²x]
= 2 / [sin²x + cos²x - cos²x]
= 2 / sin²x
= 2 csc²x
= R.H.S.

Hence, [1 / (1 + cos x)] + [1 / (1 - cos x)] = 2 csc²x


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2.
L.H.S.
= (1 - sec x) / (1 + sec x)
= [1 - (1/cos x)] / [1 + (1/cos x)]
= [(cos x/cos x) - (1/cosx)] / [(cos x/cos x) + (1/cos x)]
= [(cos x - 1) / cosx] / [(cos x + 1) / cos x]
= [(cos x - 1) / cosx] × [cos x / (cos x + 1)
= (cos x - 1) / (cos x + 1)
= R.H.S.

Hence, (1 - sec x) / (1 + sec x) = (cos x - 1) / (cos x + 1)


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3.
L.H.S.
= cos t / (1 + sin t)
= [cos t (1 - sin t)] / [(1 + sin t)(1 - sin t)]
= [cos t (1 - sin t)] / (1 - sin²t)
= [cos t (1 - sin t)] / (sin²t + cos²t - sin²t)
= [cos t (1 - sin t)] / cos²t
= (1 - sin t) / cos t
= R.H.S.

Hence, cos t / (1 + sin t) = (1 - sin t) / cos t


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4.
(1 - sin t)² / (cos²t) ≢ (1 - sin t)/ (cos t)

Maybe, it is :
L.H.S.
= (1 - sin t)² / (cos²t)
= (1 - sin t)² / (cos²t + sin²t - sin²t)
= (1 - sin t)² / (cos²t + sin²t - sin²t)
= (1 - sin t)² / (1 - sin²t)
= (1 - sin t)² / [(1 + sin t)(1 - sin t)]
= (1 - sin t) / (1 + sin t)

Hence, (1 - sin t)² / (cos²t) = (1 - sint) / (1 + sint)


====
5.
L.H.S.
= (sec x + 1) / (tan x)
= [(sec x + 1)(sec x - 1)] / [tan x (sec x - 1)]
= (sec²x - 1) / [tan x (sec x - 1)]
= tan²x / [tan x (sec x - 1)]
= tan x / (sec x - 1)
= R.H.S.

Hence, (sec x + 1) / (tan x) = tan x / (sec x - 1)

(i) (1/1 + cosx) + (1/1 - cosx) = 1 – cosx + 1 + cosx/(1 + cosx)(1 - cosx)

2/1 - cos²x (∵ (a + b) (a - b) = a² - b²)

2/sin²x (sin²x + cos²x = 1)

2cosec²x (1/sinx = cosec)



ii) (1 - secx)/ (1+secx)

(1 - 1/cosx)/ (1 + 1/cosx)

(cosx - 1)/ (cosx)/ (cosx + 1)/ cosx

(cosx - 1)/ (cosx + 1)



iii) cost/ (1 + sint)

cost/ (1 + sint)*(1 - sint)/ (1 - sint)

cost (1 - sint)/ (1 - sin²t)

cost (1 - sint)/ (cos²t) (∵ sin²x + cos²x = 1)

(1 - sint)/ (cost)



iv) (1 - sint)²/cos²t

(sin²x + cos²x = 1)

(1 - sint)²/1 - sin²t

(1 - sint)²/ (1 - sint)(1 + sint)

(1 - sint)/ (1 + sint)



v) (secx + 1)/ tanx

(secx + 1)/ tanx*(secx - 1)/ (secx - 1)

(sec²x - 1)/ tanx (secx - 1)

tan²x/ tanx (secx - 1)

tanx/ (secx - 1)

The 4th question has error. Re-check.