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Trigonometric Identities Proving Questions


(1)  Show the next identities.

(i) cotθ + tanθ  =  secθ cosecθ      Resolution

(ii) tan4θ + tan2θ  =  sec4θ – sec2θ     Resolution

(2)  Show the next identities.

(i)  (1 – tan2θ) / (cot2θ – 1)  =  tan2θ     Resolution

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

(3)  Show the next identities.

(i)  √[(1 + sinθ)/(1 – sinθ)]  =  secθ + tanθ      Resolution

(ii) [√(1 + sinθ)/(1 – sinθ)] + [√(1 – sinθ)/(1 + sinθ)]  = 2 secθ        Resolution

(4)  Show the next identities.

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

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

(5)  Show the next identities.

(i) sec4θ(1 – sin4θ) – 2tan2θ  =  1       Resolution

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

Resolution

(6)  Show the next identities.

(i) [(sinA –  sinB)/(cosA + cosB)]  + [(cosA – cosB)/(sinA + sinB)]  =  0     Resolution

(ii) [(sin3A + cos3A)/(sinA + cosA)] + [(sin3A – cos3A)/(sin A – cosA)]  =  2      Resolution

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

tanθ + cotθ  =  1     Resolution

(ii) If 3sinθ – cosθ  =  0, then present that

tan3θ  =  (3tanθ – tan3θ)/(1 – 3tan2θ)    Resolution 

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

(m2 + n2) cos2β  =  n2          Resolution

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

(x2y)2/3 – (xy2)2/3  =  1          Resolution

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

q(p2 −1)  =  2p        Resolution

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

cos6θ – 4 cos4θ + 8cos2θ  =  4        Resolution

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

(a2 – 1)/(a2 + 1)  =  sin θ          Resolution

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