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Answers
Step-by-step explanation:
LHS=
tan
2
θ−1
tan
2
θ
+
sec
2
θ−cosec
2
θ
cosec
2
θ
= \frac{\frac{sin^{2}\theta}{cos^{2}\theta}}{\frac{sin^{2}\theta}{cos^{2}\theta}-1} + \frac{\frac{1}{sin^{2}\theta}}{\frac{1}{cos^{2}\theta} - \frac{1}{sin^{2}\theta}}=
cos
2
θ
sin
2
θ
−1
cos
2
θ
sin
2
θ
+
cos
2
θ
1
−
sin
2
θ
1
sin
2
θ
1
= \frac{\frac{sin^{2}\theta}{cos^{2}\theta}}{\frac{sin^{2}\theta-cos^{2}\theta}{sin^{2}\theta cos^{2}\theta}} + \frac{\frac{1}{sin^{2}\theta}}{\frac{sin^{2}\theta-cos^{2}\theta}{sin^{2}\theta cos^{2}\theta }}=
sin
2
θcos
2
θ
sin
2
θ−cos
2
θ
cos
2
θ
sin
2
θ
+
sin
2
θcos
2
θ
sin
2
θ−cos
2
θ
sin
2
θ
1
= \frac{ sin^{2}\theta cos^{2}\theta}{cos^{2}\theta(sin^{2}\theta - cos^{2}\theta)} + \frac{ sin^{2}\theta cos^{2}\theta}{sin^{2}\theta(sin^{2}\theta-cos^{2}\theta}=
cos
2
θ(sin
2
θ−cos
2
θ)
sin
2
θcos
2
θ
+
sin
2
θ(sin
2
θ−cos
2
θ
sin
2
θcos
2
θ
= \frac{ sin^{2}\theta}{sin^{2}\theta-cos^{2}\theta} + \frac{ cos^{2}\theta}{sin^{2}\theta-cos^{2}\theta}=
sin
2
θ−cos
2
θ
sin
2
θ
+
sin
2
θ−cos
2
θ
cos
2
θ
= \frac{ sin^{2}\theta + cos^{2}\theta}{sin^{2}\theta - cos^{2}\theta}=
sin
2
θ−cos
2
θ
sin
2
θ+cos
2
θ
\begin{gathered}= \frac{1}{sin^{2}\theta - cos^{2}\theta} \\= RHS\end{gathered}
=
sin
2
θ−cos
2
θ
1
=RHS
Therefore.,
\red {\frac{tan^{2}\theta}{tan^{2}\theta-1} + \frac{cosec^{2}\theta}{sec^{2}\theta-cosec^{2}\theta}}
tan
2
θ−1
tan
2
θ
+
sec
2
θ−cosec
2
θ
cosec
2
θ
\green {= \frac{1}{sin^{2}\theta - cos^{2}\theta}}=
sin
2
θ−cos
2
θ
1