If (sec α + tan α) / (sec α − tan α) = 7/4, then find the value of cosec α.

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Given: (sec α + tan α) / (sec α − tan α) = 7/4

Step 1: Use the identity (sec α + tan α)(sec α − tan α) = 1.

Step 2: Let x = (sec α + tan α) / (sec α − tan α) = 7/4.

Step 3: Then, (sec α + tan α) = x(sec α − tan α).

Step 4: Multiply both sides by (sec α − tan α):
(sec α + tan α)(sec α − tan α) = x(sec α − tan α)^2

But (sec α + tan α)(sec α − tan α) = 1, so:
1 = x(sec α − tan α)^2

Step 5: Substitute x = 7/4:
1 = (7/4)(sec α − tan α)^2

Step 6: (sec α − tan α)^2 = 4/7

Step 7: Recall that (sec α + tan α) = 7/4 (sec α − tan α), so:
(sec α + tan α) = (7/4)(sec α − tan α)

Step 8: Let A = sec α and B = tan α.
From the above:
A + B = (7/4)(A - B)

Multiply both sides by 4:
4A + 4B = 7A - 7B

Bring terms to one side:
4A + 4B - 7A + 7B = 0

-3A + 11B = 0

11B = 3A

B = (3/11)A

Step 9: Use the identity sec^2 α - tan^2 α = 1:
A^2 - B^2 = 1

Substitute B = (3/11)A:
A^2 - (3/11)^2 A^2 = 1

A^2 (1 - 9/121) = 1

A^2 (112/121) = 1

A^2 = 121/112

Step 10: Now, A = sec α = sqrt(121/112) = 11 / sqrt(112)

Step 11: tan α = B = (3/11)A = (3/11) * (11 / sqrt(112)) = 3 / sqrt(112)

Step 12: Recall that sec α = 1 / cos α, tan α = sin α / cos α.

So cos α = 1 / sec α = sqrt(112) / 11

sin α = tan α * cos α = (3 / sqrt(112)) * (sqrt(112) / 11) = 3 / 11

Step 13: cosec α = 1 / sin α = 11 / 3

Therefore, the value of cosec α is 11/3.

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