Please give solution of example 15 Example 15: Prove that sin θ-cos θ+1sin θ+cos θ-1=1sec θ-tan θ - Maths - Introduction to Trigonometry

Question

Please give solution of example 15

Example 15: Prove that sin   θ - cos   θ + 1
sin   θ + cos   θ - 1 = 1 sec   θ -
tan   θ using the identity sec2 θ = 1 +
tan2 θ

Roopam Pandey · Accepted Answer

Dear student,

Example 15: Prove that sinθ-cosθ+1sinθ+cosθ-1=1secθ-tanθTaking LHS , divide by cosθ in numerator and denominatorsinθ-cosθ+1sinθ+cosθ-1=sinθcosθ-cosθcosθ+1cosθsinθcosθ+cosθcosθ-1cosθ                        =tanθ+secθ-1tanθ-secθ+1                             =tanθ+secθ-1tanθ-secθ+1 ×tanθ-secθtanθ-secθ                       =tanθ+secθtanθ-secθ-1×tanθ-secθtanθ-secθ+1tanθ-secθ                       =tan2θ-sec2θ-tanθ+secθtanθ-secθ+1tanθ-secθ1+tan2θ=sec2θ⇒sec2θ-tan2θ=1                      =-1-tanθ+secθtanθ-secθ+1tanθ-secθ                      =-tanθ-secθ+1tanθ-secθ+1tanθ-secθ=-1tanθ-secθ                      =-1-secθ-tanθ=1secθ-tanθ  =RHS
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Please give solution of example 15. Example 15: Prove that sin θ - cos θ + 1 sin θ + cos θ - 1 = 1 sec θ - tan θ using the identity sec2 θ = 1 + tan2 θ .

Please give solution of example 15.

Example 15: Prove that $\frac{\sin θ - \cos θ + 1}{\sin θ + \cos θ - 1} = \frac{1}{\sec θ - \tan θ}$ using the identity sec² $θ$ = 1 + tan² $θ$ .