PROVING TRIGONOMETRIC IDENTITIES WORKSHEET WITH SOLUTIONS

(1) Determine whether each of the following is an identity or not.

(i) cos²θ + sec²θ  =  2 + sinθ     

(ii) cot²θ + cosθ  =  sin²θ   

Solution

(2) Prove the following identities

(i) sec²θ + cosec²θ  =  sec²θcosec²θ        Solution

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

(iii) √(1 - sinθ)/(1 + sinθ)  =  secθ - tanθ    Solution

(iv) cosθ/(secθ - tanθ)  =  1 + sinθ        Solution

(v) √(sec²θ + cosec²θ)  =  tanθ + cotθ      Solution

(vi) (1 + cosθ - sin²θ)/(sinθ)(1 + cosθ)  =  cotθ 

Solution

(vii) secθ(1 - sinθ)(secθ + tanθ)  =  1      Solution  

(viii) sinθ/(cosecθ + cotθ)  =  1 - cosθ      Solution

(3) Prove the following identities

(i) [sin(90 - θ)/(1 + sinθ)] + [cosθ/(1 - (cos(90 - θ))]

=  2secθ       Solution

(ii) tanθ/(1 - cotθ) + cotθ/(1 - tanθ)  =  1 + secθ cosecθ  

Solution

(iii) sin(90 - θ)/(1 - tanθ) + cos(90 - θ)/(1 - cotθ) 

=  cosθ + sinθ      Solution

(iv) [tan(90 - θ)/(cosecθ + 1)] + [(cosecθ + 1)/cotθ)]

=  2secθ       Solution

(v) (cotθ + cosecθ - 1)/(cotθ - cosecθ + 1) 

=  cosecθ + cotθ    Solution

(vi) (1 + cotθ - cosecθ)(1 + tanθ + secθ)  =  2        Solution

(vii) (sinθ - cosθ + 1)/(sinθ + cosθ - 1)  =  1/(secθ-tanθ) 

Solution

(viii) tanθ/(1 - tan²θ) = sinθsin(90 - θ)/[2sin²(90 - θ) - 1]  

(ix) [1/(cosecθ - cotθ)] - (1/sinθ) 

=  [(1/sinθ)] - [1/(cosecθ + cotθ)]             Solution

(x) (cot²θ  + sec²θ)/(tan²θ + cosec²θ)

=  sinθ cosθ(tanθ + cotθ)        Solution

(4) If x = a sec θ + b tan θ and y = a tan θ + b sec θ then prove that

x² - y²  =  a² - b²          Solution

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