Question

If sintheta + cosectheta = 2 then show that sin⁷ + cosec⁷theta =?If sin + cosectheta = 2 then show that sin⁷theta + cosec⁷theta =?​

Answer 1

Solution :-

→ sinA + cosecA = 2

Putting cosecA = 1/sinA in LHS we get,

→ sinA + 1/sinA = 2

Taking LCM now,

→ (sin²A + 1)/sinA = 2

→ (sin²A + 1) = 2sinA

→ sin²A - 2sinA + 1 = 0

→ sin²A - 2*1*sinA + 1² = 0

comparing with a² - 2ab + b² = (a - b)² we get,

→ (sinA - 1)² = 0

or,

→ sinA - 1 = 0

→ sinA = 1

Therefore,

→ cosecA = 1/sinA= 1/1 = 1

Now,

→ sin⁷A + cosec⁷A

→ (1)⁷ + (1)⁷

→ 1 + 1

→ 2 (Ans.)