Trigonometric Proofs

Rules:

sec\theta =\frac { 1 }{ cos\theta } \quad \therefore \quad { sec }^{ 2 }\theta =\frac { 1 }{ { cos }^{ 2 }\theta } \\ \\ cosec\theta =\frac { 1 }{ sin\theta } \quad \therefore \quad { cosec }^{ 2 }\theta =\frac { 1 }{ { sin }^{ 2 }\theta } \\ \\ cot\theta =\frac { 1 }{ tan\theta } =\frac { 1 }{ \frac { sin\theta }{ cos\theta } } =\frac { cos\theta }{ sin\theta } \\ \\ \therefore \quad { cot }^{ 2 }\theta =\frac { 1 }{ { tan }^{ 2 }\theta } =\frac { { cos }^{ 2 }\theta }{ { sin }^{ 2 }\theta }

Transformation 1:

{ sin }^{ 2 }\theta +{ cos }^{ 2 }\theta =1\\ \\ \frac { { sin }^{ 2 }\theta }{ { sin }^{ 2 }\theta } +\frac { { cos }^{ 2 }\theta }{ { sin }^{ 2 }\theta } =\frac { 1 }{ { sin }^{ 2 }\theta } \\ \\ 1+{ cot }^{ 2 }\theta ={ cosec }^{ 2 }\theta

Transformation 2:

{ sin }^{ 2 }\theta +{ cos }^{ 2 }\theta =1\\ \\ \frac { { sin }^{ 2 }\theta }{ { cos }^{ 2 }\theta } +\frac { { cos }^{ 2 }\theta }{ { cos }^{ 2 }\theta } =\frac { 1 }{ { cos }^{ 2 }\theta } \\ \\ { tan }^{ 2 }\theta +1={ sec }^{ 2 }\theta

Other posts you may be interested in...