these are all simple exercises, you need to just substitute the values of trigonometric ratios and simply the expressions. that is all.
tan² A - Sin² A = Sin²A / Cos² A - Sin² A = Sin² A sec²A - Sin²A = Sin ² A ( sec²A - 1) = sin² A tan²A = sin² A sin²A / cos²A ====================== ii multiply the two terms to get 1+tan A+sec A+cot A+1+cotA SecA - CosecA - cosecA tan A - cosecA secA = 2 + tanA + secA +cosA/sinA + cosA/sinA *1/cosA - 1/sinA - 1/sinA*sinA/cosA - 1/sinA * 1/cosA = 2 + sinA/cosA + 1/cosA +CosA/sinA+ 1/sinA -1/sinA - 1/cosA - 1/sinA*1/cosA = 2 + sinA/cosA + cosA/sinA - 1/sinA * 1/cosA = 2 + (sin²A +cos²A - 1 ) / (sinA cosA) = 2 ====================== LHS 1/(cosecA- cotA) - 1/sin A multiply numerator and denominator with (coseA+ cotA) (cosecA + cotA)/ 1 - cosec A = cot A
RHS 1/sinA - 1/ (cosecA + cotA) multiply numerator and denominator with (cosec A - cot A)
cosec A - (cosecA - cotA) / (cosec²A - cot²A)
= cot A
LHS = RHS = cot A ========================
(cot A + tan B) / ( cot B + tan A) = = (cosA / sinA + sin B / cosB ) / (cosB/sinB + sin A/ cosA) = (cosA cosB + sinA sin B) * CosA sin B / [ (cosA cosB + sinA sinB) * sinA cos B ] = cot A tan B as the summation term cancels.
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kvnmurty
these are all simple exercises, you need to just substitute the values of trigonometric ratios and simply the expressions. that is all.
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these are all simple exercises, you need to just substitute the values of trigonometric ratios and simply the expressions. that is all.tan² A - Sin² A = Sin²A / Cos² A - Sin² A
= Sin² A sec²A - Sin²A
= Sin ² A ( sec²A - 1)
= sin² A tan²A = sin² A sin²A / cos²A
======================
ii
multiply the two terms to get
1+tan A+sec A+cot A+1+cotA SecA - CosecA - cosecA tan A - cosecA secA
= 2 + tanA + secA +cosA/sinA + cosA/sinA *1/cosA - 1/sinA - 1/sinA*sinA/cosA - 1/sinA * 1/cosA
= 2 + sinA/cosA + 1/cosA +CosA/sinA+ 1/sinA -1/sinA - 1/cosA - 1/sinA*1/cosA
= 2 + sinA/cosA + cosA/sinA - 1/sinA * 1/cosA
= 2 + (sin²A +cos²A - 1 ) / (sinA cosA)
= 2
======================
LHS 1/(cosecA- cotA) - 1/sin A
multiply numerator and denominator with (coseA+ cotA)
(cosecA + cotA)/ 1 - cosec A = cot A
RHS 1/sinA - 1/ (cosecA + cotA)
multiply numerator and denominator with (cosec A - cot A)
cosec A - (cosecA - cotA) / (cosec²A - cot²A)
= cot A
LHS = RHS = cot A
========================
(cot A + tan B) / ( cot B + tan A) =
= (cosA / sinA + sin B / cosB ) / (cosB/sinB + sin A/ cosA)
= (cosA cosB + sinA sin B) * CosA sin B / [ (cosA cosB + sinA sinB) * sinA cos B ]
= cot A tan B
as the summation term cancels.