Question

Prove the identity secxcscx(tanx+cotx)=2+tan^2x+cot^2x

1. apply the distributive property
2. apply the definitions of secant, cosecant, tangent, and cotangent
3. simplify the expressions
4. apply the definitions of secant and cosecant
5. apply the pythagorean identities
6. simply the expressions

Answer 1

sec(x)csc(x)[tan(x) + cot(x)] = 2 + tan²(x) + cot²(x)
sec(x)csc(x)[tan(x)] + sec(x)csc(x)[cot(x)] = 2 + tan²(x) + cot²(x)
sec²(x) + csc²(x) = 2 + tan²(x) + cot²(x)
sec²(x) + csc²(x) = 1 + 1 + tan²(x) + cot²(x)
sec²(x) + csc²(x) = 1 + tan²(x) + 1 + cot²(x)
sec²(x) + csc²(x) = sec²(x) + csc²(x)

Answer 2

Hello,


`sec(x)= (1)/(cos(x)) \\ cosec(x)= (1)/(sin(x)) \\ sec(x)*cosec(x)*(tg(x)+cotg(x))=(1)/(cos(x))* (1)/(sin(x))*( (sin(x))/(cos(x)) +(cos(x))/(sin(x)))\\ = (sin^2(x)+cos^2(x))/(sin^2x*cos^2x) \\ = (1)/(sin^2x*cos^2x) \\`
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`2+tg^2(x)+cotg^2(x)=2+ (sin^2x)/(cos^2x) + (cos^2x)/(sin^2x) \\ =2+ (sin^4x+cos^4x)/(sin^2x*cos^2x) \\ =(2*sin^2x*cos^2x+sin^4x+cos^4x)/(sin^2x*cos^2x) \\ = ((sin^2x+cos^2x)^2)/(sin^2x*cos^2x)} \\ = (1)/(sin^2x*cos^2x)}`