Prove the following.

(1) sec(1 – sin) (sec + tan) = (sec-secsin)(sec + tan)
= (sec-tan)(sec+tan) [secsin = sin/cos = tan]
= sec²-tan²
= 1 [1+tan² = sec²]
Hence proved.

(2) (sec + tan) (1 – sin) = [(1/cos)+(sin/cos)](1 – sin)
= [(1+sin)/cos]×(1-sin)
= (1-sin²)/cos
= cos²/cos
= cos
Hence proved.

(3) sec² + cosec² = (1/cos²) +(1/sin²)
= (sin²+cos²)/sin² cos²
= 1/ sin² cos² [sin²+cos² = 1]
= sec²× cosec²
Hence proved.

(4) cot² – tan² = (cosec²-1)-(sec²-1) [∵cot² = cosec²-1 and tan² = sec²-1]
= cosec²-1-sec²+1
= cosec²-sec²
Hence proved.

(5) tan⁴ + tan² = tan²( tan² +1)
= tan²sec² [∵tan² +1= sec²]
= (sec²-1) sec² [∵tan² = sec²-1]
= sec⁴– sec²
Hence proved.

(6)[ 1/(1- sin θ)]+[1/(1+ sinθ)] = [(1+ sinθ)+ (1-sinθ)]/ (1+ sinθ)×(1-sinθ)
= [1+ sinθ+ 1-sinθ]/ (1- sin²θ)
= 2/cos² [1- sin²θ = cos²]
= 2sec² [∵1/ cos² = sec²]
Hence proved.