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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