TRIGONOMETRIC IDENTITIES PROVING QUESTIONS

(1)  Prove the following identities.

(i) cotθ + tanθ  =  secθ cosecθ      Solution

(ii) tan4θ + tan2θ  =  sec4θ - sec2θ     Solution

(2)  Prove the following identities.

(i)  (1 - tan2θ) / (cot2θ - 1)  =  tan2θ     Solution

(ii) cosθ/(1 + sinθ)  =  secθ + tanθ     Solution

(3)  Prove the following identities.

(i)  √[(1 + sinθ)/(1 - sinθ)]  =  secθ + tanθ      Solution

(ii) [√(1 + sinθ)/(1 - sinθ)] + [√(1 - sinθ)/(1 + sinθ)]  = 2 secθ        Solution

(4)  Prove the following identities.

(i) sec6θ  =  tan6θ + 3tan2θ sec2θ + 1         Solution

(ii) (sinθ + secθ)2 + (cosθ + cosecθ)2 = 1 + (secθ + cosecθ)2         Solution

(5)  Prove the following identities.

(i) sec4θ(1 - sin4θ) - 2tan2θ  =  1       Solution

(ii)  (cotθ - cosθ)/(cotθ + cosθ)  = (cosecθ - 1)/(cosecθ + 1)

Solution

(6)  Prove the following identities.

(i) [(sinA -  sinB)/(cosA + cosB)]  + [(cosA - cosB)/(sinA + sinB)]  =  0     Solution

(ii) [(sin3A + cos3A)/(sinA + cosA)] + [(sin3A - cos3A)/(sin A - cosA)]  =  2      Solution

(7)  (i) If sinθ + cosθ  =  3 , then prove that

tanθ + cotθ  =  1     Solution

(ii) If 3sinθ - cosθ  =  0, then show that

tan3θ  =  (3tanθ - tan3θ)/(1 - 3tan2θ)    Solution 

(8) (i) If(cosα/cosβ)  =  m and (cosα/sin β) = n then prove that

(m2 + n2) cos2β  =  n2          Solution

(ii)  If cotθ + tanθ = x and secθ - cosθ = y , then prove that 

(x2y)2/3 - (xy2)2/3  =  1          Solution

(9)  (i) If sinθ + cosθ = p and secθ + cosecθ = q, then prove that

q(p2 −1)  =  2p        Solution

(ii)  If sinθ(1 + sin2θ)  =  cos2θ, then prove that

cos6θ - 4 cos4θ + 8cos2θ  =  4        Solution

(10)  If cosθ / (1 + sinθ)  =  1/a, then prove that

(a2 - 1)/(a2 + 1)  =  sin θ          Solution

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