7. Prove that :
cot2 A (sec A-1)/
1 + sin A=
sec2A
1-sin A\
1 + sec A
Answers
Question :- Prove that {cot²A(sec A -1)} / ( 1 + sin A) = sec² A {( 1 -sin A) / (1 + sec A)}
Solution :-
solving LHS,
→ {cot²A(sec A -1)} / ( 1 + sin A)
Multiply and divide by {(1 - sin A) / (1 - sin A) = 1} and {(sec A + 1) /(sec A + 1)}
→ [{cot²A(sec A -1)} / ( 1 + sin A)] * {(1 - sin A) / (1 - sin A)} * {(sec A + 1) /(sec A + 1)}
Solving Numerator now,
cot²A * (sec A -1) * (1 - sin A) * (sec A + 1)
cot²A * (sec A -1) * (sec A + 1) * (1 - sin A)
using (a + b)(a - b) = a² - b²
cot²A * (sec²A - 1) * (1 - sin A)
using (sec²A - 1) = tan²A
cot²A * tan²A * (1 - sin A)
we know that, cot²A = (1/tan²A)
(1/tan²A) * tan²A * (1 - sin A)
(1 - sin A)
Solving Denominator now,
(1 + sinA) * (1 - sinA) * (sec A + 1)
using (a + b)(a - b) = a² - b²
(1 - sin²A) * (sec A + 1)
using (1 - sin²A) = cos²A
cos²A(sec A + 1)
cos²A(1 + sec A)
using cos²A = (1/sec²A) now,
(1/sec²A)(1 + secA)
therefore, we get,
→ (1 - sinA) /{(1/sec²A)(1 + secA)}
→ sec²A(1 - sin A) / (1 + sec A) = RHS . (Proved.)
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